< Topology without Tears

01-08-2020

Topological Spaces

Topology

1.1.1 Definition

Let \(X\) be a non-empty set. A set \(\tau\) of subsets of \(X\) is said to be a topology on \(X\) if

  1. \(X \in \tau\) and \(\emptyset \in \tau\)
  2. the union of any (infinite or finite) number of sets in \(\tau\) belongs to \(\tau\), and
  3. the intersection of any two sets in \(\tau\) belongs to \(\tau\).

The pair \((X, \tau)\) is called a topological space.

for example, Let \(\mathbb{N}\) be the set of all natural numbers, let \(\tau\) consist of \(\mathbb{N}\), \(\emptyset\), and all finite subsets of \(\mathbb{N}\). Then \(\tau\) is not a topology on \(\mathbb{N}\) as any infinite union of sets in \(\tau\) is not belonged to \(\tau\).

1.1.2 Definition

Let \(X\) be any non-empty set and let \(\tau\) be the collection of all subsets of \(X\), then \(\tau\) is called the discrete topology on \(X\). The topology space \((X, \tau)\) is called a discrete space.

There will be infinite number of discrete spaces.

If every infinite subset of an infinite subset is open or all infinite subsets are closed, then \(\tau\) must be the discrete topology.

1.1.3 Definition

Let \(X\) be any non-empty set and \(\tau = \{X, \emptyset\}\). Then \(\tau\) is called the indiscrete topology and \((X, \tau)\) is said to be an indiscrete space.

There are also infinite number of indiscrete spaces.

1.1.4 Proposition

If \((X, \tau)\) is a topological space such that, for every \(x \in X\), the singletion set \(\{x\}\) is in \(\tau\), then \(\tau\) is a discrete topology.

We can see that if \(S\) be any subset of \(X\), then

\[S = \cup_{x \in S} x\]

Since \(S\) is in \(\tau\), then \(\tau\) is the set of all subsets of \(X\).

Moreover, if all infinite subsets of an infinite set \(X\) is in topology \(\tau\), then \(\tau\) must be a discrete topology, the proof is related to open and close sets.

Open Sets, Closed Sets, and Clopen Sets

1.2.1 Definition

Let \((X, \tau)\) be any topological space. Then the members of \(\tau\) are said to be open sets.

1.2.2 Proposition

If \((X, \tau)\) is any topological space, then

  1. \(X\) and \(\emptyset\) are open sets,
  2. the union of any (finite or infinite) number of open sets is an open set, and
  3. the intersection of any finite number of open sets is an open set.

We can find that infinite intersections of open sets may not be open.

1.2.3 Definition

Let \((X, \tau)\) be a topological space. A subset \(S\) of \(X\) is said to be a closed set in \((X, \tau)\) if its complement in \(X\), namely \(X / S\), is open in \((X, \tau)\).

1.2.4 Proposition

If \((X, \tau)\) is any topological space, then

  1. \(\emptyset\) and \(X\) are closed sets,
  2. the intersection of any (finite or infinite) number of closed sets is a closed set and
  3. the union of any finite number of closed sets is a closed set.

A set can be either open or closed or open and closed or neither open nor closed.

1.2.5 Definition

A subset \(S\) of a topological space \((X, \tau)\) is said to be clopen if it is both open and closed in \((X, \tau)\).

The Finite-Closed Topology

Sometimes it is more natural to define topology by saying which sets are closed.

1.3.1 Definition

Let \(X\) be any non-empty set. A topology \(\tau\) on \(X\) is called the finite-closed topology or the cofinite topology if the closed subsets of \(X\) are \(X\) and all finite subsets of \(X\).

While all finite sets are closed, not all infinite sets are open. If a topological space has at least 3 distinct clopen subsets, then \(X\) must be a finite set.

1.3.2 Definition

Let \(f\) be a function from a set \(X\) into a set \(Y\).

  1. The function \(f\) is said to be injective if \(f(x1) = f(x2)\) implies \(x1 = x2\), for \(x1, x2 \in X\);
  2. The function \(f\) is said to be surjective if for each \(y \in Y\) there exists an \(x \in X\) such that \(f(x) = y\);
  3. The function \(f\) is said to be bijective if it is both injective and surjective.
1.3.3 Definition

Let \(f\) be a function from a set \(X\) into a set \(Y\). The function \(f\) is said to have an inverse if there exists a function \(g\) of \(Y\) into \(X\) such that \(g(f(x)) = x\), for all \(x \in X\) and \(f(g(y)) = y\), for all \(y \in Y\). The function \(g\) is called an inverse function of \(f\).

1.3.4 Proposition

Let \(f\) be a function from a set \(X\) into a set \(Y\).

  1. The function \(f\) has an inverse if and only if \(f\) is bijective.
  2. Inverse is unique.
  3. \(g\) is an inverse function of \(f\) if and only if \(f\) is an inverse function of \(g\).
1.3.5 Definition

Let \(f\) be a function from a set \(X\) into a set \(Y\). If \(S\) is any subset of \(Y\), then the set \(f^{-1}(S)\) is defined by

\[f^{-1}(S) = \{x : x \in X \land f(x) \in S\}.\]

The subset \(f^{-1}(S)\) of \(X\) is said to be the inverse image of \(S\).

There’s an interesting conclusion of topology and inverse image: Let \((Y, \tau)\) be a topological space and \(X\) a non-empty set. Further, let \(f\) be a function from \(X\) into \(Y\). Put \(\tau_{1} = \{f^{-1}(S) : S \in \tau\}\). Then \(\tau_{1}\) is a topology on \(X\).

1.3.6 Definition

A topological space \((X,\tau)\) is said to be a \(T_1\)-space if every singleton set \(\{x\}\) is closed in \((X, \tau)\).

A discrete space or an infinite set with the finite-set topology is a \(T_{1}\)-space.

A topological space is said to be a \(T_0\)-space if for each pair of distinct points \(a, b\) in \(X\), either there exists an open set containing \(a\) and not \(b\), or there exists an open set containing \(b\) and not \(a\).

Every \(T_1\)-space is a \(T_0\)-space.

The Euclidean Topology

The Euclidean Topology on \(\mathbb{R}\)

2.1.1 Definition

A subset \(S\) of \(\mathbb{R}\) is said to be open in the Euclidean topology on \(\mathbb{R}\) if it has the following property:

We have following interesting collaries from this definition.

  1. Let \(r, s \in \mathbb{R}\) with \(r < s\). In the Euclidean topology \(\tau\) on \(\mathbb{R}\), \((r, s)\) is an open set.
  2. \((r, \infty)\) and \((-\infty, r)\) are open sets for every real number \(r\).
  3. Not all open sets in Euclidean space are intervals.
  4. For each \(c\) and \(d\) in R with \(c < d\), \([c, d]\) is not an open set in \(\mathbb{R}\).
  5. For each \(a\) and \(b\) in \(\mathbb{R}\) with \(a < b\), \([a, b]\) is a closed set in the Euclidean topology on \(\mathbb{R}\).
  6. \(\mathbb{R}\) is a \(T_1\)-space.
  7. \(\mathbb{Z}\) is a closed subset of \(\mathbb{R}\).
  8. \(\mathbb{Q}\) is neither a closed subset nor an open subset of \(\mathbb{R}\).
  9. The only clopen subsets of \(\mathbb{R}\) are only \(\mathbb{R}\) and \(\emptyset\).

Basis for a Topology

2.2.1 Proposition

A subset \(S\) of \(\mathbb{R}\) is open if and only if it is a union of open intervals.

2.2.2 Definition

Let \((X, \tau)\) be a topological space. A collection \(B\) of open subsets of \(X\) is said to be a basis for the topology \(\tau\) if every open set is a union of members of \(B\).

\(B\) generates the whole topological space like basis in vector space in linear algebra.

\(\tau\) itself is a basis for \(\tau\). We do have many bases for a topology. Generally, if \(B\) is a basis for \(\tau\), then let \(B_1\) is a collection of subsets of \(X\) such that \(B \subseteq B_1 \subseteq \tau\), then \(B_1\) is also a basis for \(\tau\).

2.2.3 Proposition

Let \(X\) be a non-empty set and let \(B\) be a collection of subsets of \(X\). Then \(B\) is a basis for a topology on \(X\) if and only if \(B\) has the following properties:

  1. \(X = \cup_{b \in B} b\) and
  2. for any \(b_1, b_2 \in B\), the set \(b_1 \cap b_2\) is a union of members of \(B\).

Generally, we can define a Euclidean topology on \(\mathbb{R}^{n} = \{<x_1, x_2, ..., x_n>\ : x_i \in \mathbb{R}, i = 1, 2, ..., n\}\) as \(B = \{<x_1, x_2, ..., x_n>\ \in \mathbb{R}^{n} : a_i < x_i < b_i, i = 1, 2, ..., n\}\) of \(\mathbb{R}^{n}\) with sides parallel to the axes.

In fact, we will often use open discs as a group of basis since it it the expansion of intervals on \(\mathbb{R}\).

2.2.4 Definition

A topological space \((X, \tau)\) is said to satisfy the second axiom of countability if there exists a basis \(B\) for \(\tau\), where \(B\) consists of only a countable number of sets.

Basis for a Given Topology

2.3.1 Proposition

Let \((X, \tau)\) be a topological space. A family \(B\) of open subsets of \(X\) is a basis for \(\tau\) if and only if for any point \(x\) belonging to any open set \(U\), there is \(b \in B\) such that \(x \in b \subseteq U\).

2.3.2 Proposition

Let \(B\) be a basis for a topology \(\tau\) on a set \(X\). Then a subset \(U\) of \(X\) is open if and only if for each \(x \in U\) there exists a \(b \in B\) such that \(x \in b \subseteq U\).

2.3.3 Proposition

Let \(B_1\) and \(B_2\) be bases for topologies \(\tau_1\) and \(\tau_2\), respectively, on a non-empty set \(X\). Then \(\tau_1 = \tau_2\) if and only if

  1. for each \(b \in B_{1}\) and each \(x \in b\), there exists a \(b' \in B_{2}\) such that \(x \in b' \subseteq b\), and
  2. for each \(b \in B_{2}\) and each \(x \in b\), there exists a \(b' \in B_{1}\) such that \(x \in b' \subseteq b\).
2.3.4 Definition

Let \((X, \tau)\) be a topological space. A non-empty collection \(S\) of open subsets of \(X\) is said to be a subbasis for \(\tau\) if the collection of all finite intersections of members of \(S\) forms a basis for \(\tau\).

Limit Points

Limit Points and Closure

It is better to say the elements of set \(X\) as point if \((X, \tau)\) is a topological space.

3.1.1 Definition

Let \(A\) be a subset of a topological space \((X, \tau)\). A point \(x \in X\) is said to be a limit point (or accumulation point or cluster point) of \(A\) if every open set \(U\), containing \(x\) contains a point of \(A\) different from \(x\).

It is easy to verify that discrete space has no limit point. In indiscrete space, a set with at least two point will have all \(x \in X\) as its limit points.

3.1.2 Proposition

Let \(A\) be a subset of a topological space \((X, \tau)\). Then \(A\) is closed in \((X, \tau)\) if and only if \(A\) contains all of its limit points.

3.1.3 Proposition

Let \(A\) be a subset of a topological space \((X, \tau)\) and \(A'\) the set of all limit points of \(A\). Then \(A \cup A'\) is a closed set.

3.1.4 Definition

Let \(A\) be a subset of a topological space \((X, \tau)\). Then the set \(A \cup A'\) consisting of \(A\) and all its limit points is called the closure of \(A\) and is denoted by \(\bar{A}\).

\(\bar{A}\) is the smallest closed set containing \(A\), this implies that \(\bar{A}\) is the intersection of all closed sets containing \(A\).

For example, the closure of \(\mathbb{Q}\) on \(\mathbb{R}\) is \(\mathbb{R}\) as every interval must contain one rational number.

3.1.5 Definition

Let \(A\) be a subset of a topological space \((X, \tau)\). Then \(A\) is said to be dense in \(X\) or everywhere dense in \(X\) if \(\bar{A} = X\).

So \(\mathbb{Q}\) is a dense subset of \(\mathbb{R}\).

3.1.6 Proposition

Let \(A\) be a subset of a topological space \((X, \tau)\). Then \(A\) is dense in \(X\) if and only if every non-empty open subset of \(X\) intersects \(A\) non-trivially (that is, if \(U \in \tau\) and \(U \neq \emptyset\) then \(A \cap U \neq \emptyset\)).

Neighborhoods

3.2.1 Definition

Let \((X, \tau)\) be a topological space, \(N\) a subset of \(X\) and \(p\) a point in \(N\). Then \(N\) is said to be a neighborhood of the point if there exists an open set \(U\) such that \(p \in U \subseteq N\).

3.2.2 Proposition

Let \(A\) be a subset of a topological space \((X, \tau)\). A point \(x \in X\) is a limit point of \(A\) if and only if every neighborhood of \(x\) contains a point of \(A\) different from \(x\).

3.2.3 Corollary

Let \(U\) be a subset of a topological space \((X, \tau)\). Then \(U \in \tau\) if and only if for each \(x \in U\) there exists a \(V \in \tau\) such that \(x \in V \subseteq U\).

3.2.4 Definition

A topological space \((X, \tau)\) is said to be separable if it has a dense subset which is countable.

Connectedness

3.3.1 Lemma

Let \(S\) be a subset of \(\mathbb{R}\) which is bounded above and let \(p\) be the supremum of \(S\). If \(S\) is a closed subset of \(\mathbb{R}\), then \(p \in S\).

3.3.2 Proposition

Let \(T\) be a clopen subset of \(\mathbb{R}\). Then either \(T = \mathbb{R}\) or \(T = \emptyset\).

3.3.3 Definition

Let \((X, \tau)\) be a topological space. Then it is said to be connected if the only clopen subsets of \(X\) are \(X\) and \(\emptyset\).

\(\mathbb{R}\) is connected as above.

3.3.4 Proposition

A topological space \((X, \tau)\) is disconnected if and only if there are non-empty open sets \(A\) and \(B\) such that \(A \cap B = \emptyset\) and \(A \cup B = X\).

Homeomorphisms

Subspaces

4.1.1 Definition

Let \(Y\) be a non-empty subset of a topological space \((X, \tau)\). The collection \(\tau_Y = \{O \cap Y : O \in \tau\}\) of subsets of \(Y\) is a topology on \(Y\) called the subspace topology (or the relative topology or the induced topology or the topology induced on \(Y\) by \(\tau\)). The topological space \((Y, \tau_Y)\) is said to be a subspace of \((X, \tau)\).

Since the subspace will change the sets’ properties, we must make clear in what space or what topology when we are talking about properties like open and close.

Topology induced on \(\mathbb{Z}\) by Euclidean topology on \(\mathbb{R}\) is a discrete topology.

4.1.2 Definition

A topological space \((X, \tau)\) is said to be Hausdorff (or a \(T_2\)-space) if given any pair of distinct points \(a, b \in X\) there exists open sets \(U\) and \(V\) such that \(a \in U\), \(b \in V\), and \(U \cap V = \emptyset\).

every \(T_2\)-space is a \(T_1\)-space.

\(\mathbb{R}\) is a \(T_2\)-space, and any subspace of \(T_2\)-space is a \(T_2\)-space.

4.1.3 Definition

A topological space \((X, \tau)\) is said to be a regular space if for any closed subset \(A\) of \(X\) and any point \(x \in X / A\), there exist open sets \(U\) and \(V\) such that \(x \in U\), \(A \subseteq V\), and \(U \cap V = \emptyset\). If \((X, \tau)\) is regular and a \(T_1\)-space, then it is said to be a \(T_3\)-space.

Homeomorphisms

4.2.1 Definition

Let \((X, \tau)\) and \((Y, \tau_1)\) be topological spaces. Then they are said to be homeomorphic if there exists a function \(f: X \to Y\) which has the following properties:

  1. \(f\) is a bijection
  2. for each \(U \in \tau_1\), \(f^{-1}(U) \in \tau\), and
  3. for each \(V \in \tau\), \(f(V) \in \tau_1\).

Further, the map \(f\) is said to be a homeomorphism between \((X, \tau)\) and \((Y, \tau_1)\). We write \((X, \tau) \equiv (Y, \tau_1)\).

Homeomorphism is a equivalence relation.

Every open interval \((a, b)\) is homeomorphic to \(\mathbb{R}\).

Non-Homeomorphic Spaces

4.3.1 Proposition

Any topological space homeomorphic to a connected space is connected.

4.3.2 Remark

So far we have known twelve properties preserved by homeomorphisms, they are:

  1. \(T_0\)-space, \(T_1\)-space, \(T_2\)-space, regular space and \(T_3\)-space
  2. the second axiom of countability
  3. separable space
  4. discrete and indiscrete space
  5. cofinite and cocountable space
  6. connectedness

If we want to prove two spaces are not homeomorphic, we just need to check whether one meets these properties and another not.

Also, homeomorphism must have the same cardinality on set and topology.

4.3.3 Definition

A subset of \(S\) of \(\mathbb{R}\) is said to be an interval if it has the following property: if \(x \in S\), \(z \in S\), and \(y \in \mathbb{R}\) are such that \(x \lt y \lt z\), then \(y \in S\).

4.3.4 Proposition

A subspace \(S\) of \(\mathbb{R}\) is connected if and only if it is an interval.

4.3.5 Corollary

If \(a, b, c,\) and \(d\) are real numbers with \(a \lt b\) and \(c \lt d\), then

  1. \((a, b) \not\equiv [c, d)\),
  2. \((a, b) \not\equiv [c, d]\), and
  3. \([a, b) \not\equiv[c, d]\).

Continuous Mappings

Continuous Mappings

5.1.1 Lemma

Let \(f\) be a function mapping \(\mathbb{R}\) into itself. Then \(f\) is continuous if and only if for each \(a \in \mathbb{R}\) and each open set \(U\) containing \(f(a)\), there exists an open set \(V\) containing \(a\) such that \(f(V) \subseteq U\).

5.1.2 Lemma

Let \(f\) be a mapping of a topological space \((X, \tau)\) into a topological space \((Y, \tau')\). Then the following two conditions are equivalent:

  1. for each \(U \in \tau'\), \(f^{-1}(U) \in \tau\);
  2. for each \(a \in X\) and each \(U \in \tau'\) with \(f(a) \in U\), there exists a \(V \in \tau\) such that \(a \in V\) and \(f(V) \subseteq U\).
5.1.3 Definition

Let \((X, \tau)\) and \((Y, \tau_1)\) be topological spaces and \(f\) a function from \(X\) into \(Y\). Then \(f : (X, \tau) \to (Y, \tau_1)\) is said to be a continuous mapping if for each \(U \in \tau_1\), \(f^{-1}(U) \in \tau\).

Continuous relation is transitive.

If \(f\) and its inverse are all continuous, then \(f\) is a homeomorphism.

5.1.4 Proposition

Let \((X, \tau\)) and \((Y, \tau_1)\) be topological spaces, \(f : (X, \tau) \to (Y, \tau_1)\) a continuous mapping. \(A\) a subset of \(X\), and \(\tau_2\) the induced topology on \(A\). Further let \(g : (A, \tau_2) \to (Y, \tau_1)\) be the restriction of \(f\) to \(A\); that is, \(g(x) = f(x)\), for all \(x \in A\). Then \(g\) is continuous.

Intermediate Value Theorem

5.2.1 Proposition

Let \((X, \tau)\) and \((Y, \tau1)\) be topological spaces and \(f : (X, \tau) \to (Y, \tau_1)\) surjective and continuous. If \((X, \tau)\) is connected, then \((Y, \tau_1)\) is connected.

5.2.2 Definition

A topological space \((X, \tau)\) is said to be path-connected (or pathwise connected) if for each pair of (distinct) points \(a\) and \(b\) of \(X\) there exists a continuous mapping \(f : [0,1] \to (X, \tau)\), such that \(f(0) = a\) and \(f(1) = b\). The mapping \(f\) is said to be a path joining \(a\) to \(b\).

Every interval and \(\mathbb{R}^{n}\) is path connected.

5.2.3 Proposition

Every path-connected space is connected.

5.2.4 Theorem(Weierstrass Intermediate Value Theorem)

Let \(f : [a,b] \to \mathbb{R}\) be continuous and let \(f(a) \neq f(b)\). Then for every number \(p\) between \(f(a)\) and \(f(b)\) there is a point \(c \in [a,b]\) such that \(f(c) = p\).

5.2.5 Corollary(Fixed Point Theorem)

Let \(f\) be a continuous mapping of \([0,1]\) into \([0,1]\). Then there exists a \(z \in [0,1]\) such that \(f(z) = z\). \(z\) is a fixed point.

This is a special case of Brouwer Fixed Point Theorem.

Metric Spaces

Metric Spaces

6.1.1 Definition

Let \(X\) be a non-empty set and \(d\) a real-valued function defined on \(X \times X\) such that for \(a, b \in X\):

  1. \(d(a, b) \ge 0\) and \(d(a, b) = 0\) if and only if \(a = b\);
  2. \(d(a, b) = d(b, a)\); and
  3. \(d(a, c) \le d(a, b) + d(b, c)\), for all \(a, b\) and \(c\) in \(X\) [the triangle inequality]

Then \(d\) is said to be a metric on \(X\), \((X, d)\) is called a metric-space and \(d(a, b)\) is referred to as the distance between \(a\) and \(b\).

We have \(\|a - b\|\) for \(\mathbb{R}\) and \(\sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2}\) for \(\mathbb{R}^2\), which is also called the euclidean-metric.

6.1.2 Remark

We often define metrics on #function-spaces. For example, let \(\subset[0, 1]\) be the set of all continuous functions from \([0, 1]\) into \(\mathbb{R}\), we have many metrics like:

  1. \[d(f, g) = \int_{0}^{1}|f(x) - g(x)|dx\]
  2. \[d^{*}(f, g) = sup\{|f(x) - g(x)| : x \in [0, 1]\}\]
6.1.3 Definition

Let \((X, d)\) be a metric space and \(r\) any positive real number. Then the #open-ball about \(a \in X\) of radius \(r\) is the set \(B_{r}(a) = \{x : x \in X \land d(a, x) < r\}\).

6.1.4 Lemma

Let \((X, d)\) be a metric space and \(a\) and \(b\) points of \(X\). Further, let \(\delta_{1}\) and \(\delta_{2}\) be positive real numbers. If \(c \in B_{\delta_1}(a) \cap B_{\delta_2}(b)\), then there exists a \(\delta < 0\) such that \(B_{\delta}(c) \subseteq B_{\delta_1}(a) \cap B_{\delta_2}(b)\).

That is to say, any intersection of two open balls is a union of open balls.

6.1.5 Proposition

Let \((X, d)\) be a metric space. Then the collection of open balls in \((X, d)\) is a basis for a topology \(\tau\) on \(X\).

Not only euclidean metric induces euclidean topology.

6.1.6 Definition

Metrics on a set \(X\) are said to be #equivalent if they induce the same topology on \(X\).

6.1.7 Proposition

Let \((X, d)\) be a metric space and \(\tau\) the topology induced on \(X\) by the metric \(d\). Then a subset \(U\) of \(X\) is open in \((X, \tau)\) if and only if for each \(a \in U\) there exists an \(\epsilon > 0\) such that the open ball \(B_{\epsilon}(a) \subseteq U\).

6.1.8 Proposition

Let \((X, d)\) be any metric space and \(\tau\) the topology induced on \(X\) by \(d\). Then \((X, \tau)\) is a Hausdorff space.

6.1.9 Definition

A space \((X, \tau)\) is said to be #metrizable if there exists a metric \(d\) on the set \(X\) with the property that \(\tau\) is the topology induced by \(d\).

6.1.10 Definition

A topological space \((X, \tau)\) is said to be a #normal-space if for each pair of disjoint closed sets \(A\) and \(B\), there exists open sets \(U\) and \(V\) such that \(A \subseteq U\), \(B \subseteq V\), and \(U \cap V = \emptyset\).

A normal Hausdorff space is called a \(T_4\) -space.

6.1.11 Definition

Let \((X, d)\) and \((Y, d_1)\) be metric spaces. Then \((X, d)\) is said to be #isometric to \((Y, d_1)\) if there exists a surjective mapping \(f : (X, d) \to (Y, d_1)\) such that for all \(x_1\) and \(x_2\) in \(X\),

\(d(x_1, x_2) = d_1(f(x_1), f(x_2))\).

Such a mapping \(f\) is said to be an #isometry.

Isometric metric spaces are homeomorphic.

6.1.12 Definition

A topological space \((X, \tau)\) is said to satisfy the #first-axiom-of-countability or be #first-countable if for each \(x \in X\) there exists a countable family \(\{ U_i(x) \}\) of open sets containing \(x\) with the property that every open set \(V\) containing \(x\) has (at least) one of the \(U_i(x)\) as a subset. The countable family \(\{ U_i(x) \}\) is said to be a #countable-base at \(x\).

Every metrizable space satisfies the first axiom of countability.

Every topological space satisfying the second axiom of countability satisfies the first axiom of countability.

6.1.13 Definition

A topological space \((X, \tau)\) is said to be #locally-euclidean if there exists a positive integer \(n\) such that each point \(x \in X\) has an open neighbourhood homeomorphic to an open ball about 0 in \(\mathbb{R}^n\) with the euclidean metric. A Hausdorff locally euclidean space is said to be a #topological-manifold.

Convergence of Sequences

6.2.1 Definition

Let \((X, d)\) be a metric space and \(x_1, x_2, ... , x_n, ...\) a sequence of points in \(X\). Then the sequence is said to be #converge-to \(x \in X\) if given any \(\epsilon > 0\) there exists an integer \(n_0\) such that for all \(n \ge n_0\), \(d(x, x_n) < \epsilon\) . This is denoted by \(x_n \to x\).

The sequence is said to be #convergent.

6.2.2 Proposition

Let \(x_1, x_2, ... , x_n, ...\) be a sequence of points in a metric space \((X, d)\). Further, let \(x\) and \(y\) be points in \((X, d)\) such that \(x_n \to x\) and \(x_n \to y\). Then \(x = y\).

6.2.3 Proposition

Let \((X, d)\) be a metric space. A subset \(A\) of \(X\) is closed in \((X, d)\) if and only if every convergent sequence of points in \(A\) converges to a point in \(A\).

6.2.4 Proposition

Let \((X, d)\) and \((Y, d_1)\) be metric spaces and \(f\) a mapping of \(X\) into \(Y\). Let \(\tau\) and \(\tau_1\) be the topologies determined by \(d\) and \(d_1\), respectively. Then \(f: (X, \tau) \to (Y, \tau_1)\) is continuous if and only if \(x_n \to x \implies f(x_n) \to f(x)\); that is, if \(x_1, x_2, ... , x_n, ...\) is a sequence of points in \((X, d)\) converging to \(x\), then the sequence of points \(f(x_1), f(x_2), ... , f(x_n), ...\) in \((Y, d_1)\) converges to \(f(x)\).

6.2.5 Definition

A topological space \((X,\tau)\) is said to be a #sequential-space if every sequentially closed set is closed.

6.2.6 Definition

A topological space \((X, \tau)\) is said to be #Frechet-Uryson-space if for every subset \(S\) of \((X, \tau)\) and every \(a\) in the closure \(\overline{S}\), of \(S\) there is a sequence \(s_n \to a\), for \(s_n \in S\), \(n \in \mathbb{N}\).

6.2.7 Definition

A topological space \((X, \tau)\) is said to have #countable-tightness if for each subset \(S\) of \(X\) and each \(x \in \overline{S}\), there exists a countable set \(C \subseteq S\), such that \(x \in \overline{C}\).

Implication: Metrizable -> First Countable -> Frechet-Urysohn -> Sequential -> Countable Tightness

Completeness

6.3.1 Definition

A sequence \(x_1, x_2, ..., x_n,...\) of points in a metric space \((X, d)\) is said to be a #Cauchy-sequence if given any real number \(\epsilon > 0\), there exists a positive integer \(n_0\), such that for all integers \(m \ge n_0\) and \(n \ge n_0\), \(d(x_m, x_n) < \epsilon\).

6.3.2 Proposition

Let \((X, d)\) be a metric space and \(x_1 ,x_2, ..., x_n,...\) a sequence of points in \((X, d)\). If there exists a point \(a \in X\), such that \(x_n \to a\), then the sequence is a Cauchy sequence.

But a Cauchy sequence does not have to be a convergent sequence.

6.3.3 Definition

A metric space \((X, d)\) is said to be #complete if every Cauchy sequence in \((X, d)\) converges to a point in \((X, d)\).

Now it’s time to show \(\mathbb{R}\) is a complete metric space.

6.3.4 Definition

If \(\{x_n\}\) is any sequence, then the sequence \(x_{n_1}, x_{n_2}, ...\) is said to be a #subsequence if \(n_1 < n_2 < n_3 < ...\).

6.3.5 Definition

Let \(\{x_n\}\) be a sequence in \(\mathbb{R}\). Then it is said to be an #increasing-sequence if \(x_n \le x_{n+1}\), for all \(n \in \mathbb{N}\). It is said to be a #decreasing-sequence if \(x_n \ge x_{n+1}\), for all \(n \in \mathbb{N}\). A sequence which is either increasing or decreasing is said to be #monotonic.

6.3.6 Definition

Let \(\{x_n\}\) be a sequence in \(\mathbb{R}\). Then \(n_0 \in \mathbb{N}\) is said to be a #peak-point if \(x_n \le x_{n_0}\), for every \(n \ge n_0\).

6.3.7 Lemma

Let \(\{x_n\}\) be any sequence in \(\mathbb{R}\). Then \(\{x_n\}\) has a monotonic subsequence.

6.3.8 Proposition

Let \(\{x_n\}\) be a monotonic sequence in \(\mathbb{R}\) with the euclidean metric. Then \(\{x_n\}\) converges to a point in \(\mathbb{R}\) if and only if \(\{ x_n \}\) is bounded.

6.3.9 Theorem(Bolzano-Weierstrass Theorem)

Every bounded sequence in \(\mathbb{R}\) with the euclidean metric has a convergent subsequence.

6.3.10 Corollary

The metric space \(\mathbb{R}\) with the euclidean metric is a complete metric space.

First we show that all Cauchy sequences are bounded, by 6.3.9, we know there’s a subsequence convergent to \(a\). Finally, we show the whole Cauchy sequence is convergent to \(a\).

The metric space \(\mathbb{R}^n\) with the euclidean metric is a complete metric space.

6.3.11 Proposition

Let \((X, d)\) be a metric space, \(Y\) a subset of \(X\), and \(d_1\) the metric induced on \(Y\) by \(d\).

  1. If \((X, d)\) is complete and \(Y\) a closed subspace of \((X, d)\), then \((Y, d_1)\) is complete.
  2. If \((Y, d_1)\) is complete, then \(Y\) is a closed subspace of \((X, d)\).
6.3.12 Definition

A topological space \((X, \tau)\) is said to be #completely-metrizable if there exists a metric \(d\) on \(X\) such that \(\tau\) is the topology on \(X\) determined by \(d\) and \((X, d)\) is a complete metric space.

We see that completeness is not a topological property but completely metrizable is.

6.3.13 Definition

A topological space \((X, \tau)\) is said to be a #Polish-space if it is separable and completely metrizable.

6.3.14 Definition

A topological space \((X, \tau)\) is said to be a #Souslin-space if it is Hausdorff and a continuous image of a Polish space. If \(A\) is a subset of a topological space \((Y, \tau_1)\) such that with the induced topology \(\tau_2\), the space \((A, \tau_2)\) is a Souslin space, then \(A\) is said to be an #analytic-set in \((Y, \tau_1)\).

6.3.15 Definition

Let \((X, d)\) and \((Y, d_1)\) be metric spaces and \(f\) a mapping of \(X\) into \(Y\). Let \(Z = f(X)\), and \(d_2\) be the metric induced on \(Z\) by \(d_1\). If \(f:(X, d) \to (Z, d_2)\) is an isometry, then \(f\) is said to be an #isometric-embedding of \((X, d)\) in \((Y, d_1)\).

\(\mathbb{N}\) has an natural embedding to \(\mathbb{Q}\) and \(\mathbb{R}\).

6.3.16 Definition

Let \((X, d)\) and \((Y, d_1)\) be metric spaces and \(f\) a mapping of \(X\) into \(Y\). If \((Y, d_1)\) is a complete metric space, \(f: (X, d) \to (Y, d_1)\) is an isometric embedding and \(f(X)\) is a dense subset of \(Y\) in the associated topological space, then \((Y, d_1)\) is said to be a #completion of \((X, d)\).

6.3.17 Proposition

If \((X, d)\) is a metric space, then it has a completion.

6.3.18 Proposition

Isometric metric spaces’ completions are also isometric.

6.3.19 Definition

Let \((N, \Vert \Vert)\) be a normed vector space and \(d\) the associated metric on the set \(N\). Then \((N, \vert \vert)\) is said to be a #Banach-space if \((N, d)\) is a complete metric space.

Contraction Mappings

6.4.1 Definition

Let \((X, d)\) be a metric space and \(f\) a mapping of \(X\) into itself. Then \(f\) is said to be a #contraction-mapping if there exists an \(r \in (0, 1)\), such that

\(d(f(x_1), f(x_2)) \le r.d(x_1, x_2)\), for all \(x_1, x_2 \in X\).

6.4.2 Proposition

Let \(f\) be a contraction mapping of the metric space \((X, d)\). Then \(f\) is a continuous mapping.

6.4.3 Theorem(Banach Fixed Point Theorem)

Let \((X, d)\) be a complete metric space and \(f\) a contraction mapping of \((X, d)\) into itself. Then \(f\) has precisely one fixed point.

Generally, A function with its Nth derivate a contraction mapping has also only one fixed point.

Baire Spaces

6.5.1 Theorem(Baire Category Theorem)

Let \((X, d)\) be a complete metric space. If \(X_1, X_2, ... , X_n, ...\) is a sequence of open dense subsets of \(X\), then the set \(\cap_{n=1}^{\infty}X_n\) is also dense in \(X\).

6.5.2 Definition

Let \((X, \tau)\) be any topological space and \(A\) any subset of \(X\). The largest open set contained in \(A\) is called the #interior of \(A\) and is denoted by \(Int(A)\). Each point \(x \in Int(A)\) is called an #interior-point of \(A\). The set \(Int(X \setminus A)\), denoted by \(Ext(A)\), and is called the #exterior of \(A\) and each point in \(Ext(A)\) is called an #exterior-point of \(A\). The set \(\overline{A} \setminus Int(A)\) is called the #boundary of \(A\). Each point in the boundary of \(A\) is called a #boundary-point of \(A\).

6.5.3 Definition

A subset \(A\) of a topological space \((X, \tau)\) is said to be #nowhere-dense if the set \(\overline{A}\) has empty interior.

Therefore 6.5.1 can be expressed like: there is at least one \(X_n\) in sequence of dense subsets that the set \(\overline{X}\) is not nowhere dense.

6.5.4 Definition

A topological space \((X, d)\) is said to be a #Baire-space if for every sequence \(\{X_n\}\) of open dense subsets of \(X\), the set \(\cap_{n=1}^{\infty}X_n\) is also dense in \(X\).

So Every complete metrizable space is a Baire space.

\(\mathbb{Q}\) is not a Baire space.

6.5.5 Definition

Let \(Y\) be a subset of a topological space \((X, \tau)\). If \(Y\) is a union of a countable number of nowhere dense subsets of \(X\), then \(Y\) is said to be a set of the #first-category or #meager in \((X, \tau)\). If \(Y\) is not first category, it is said to be a set of the #second-category in \((X, \tau)\).

6.5.6 Proposition

If \(Y\) is a first category subset of a Baire space \((X, \tau)\), then the interior of \(Y\) is empty.

Combining 6.5.1 and 6.5.6, we know that \(X \setminus Y\) is a second category set.

6.5.7 Definition

Let \(S\) be a subset of a real vector space \(V\). The set \(S\) is said to be #convex if for each \(x, y \in S\) and every real number \(0 < \lambda < 1\), the point \(\lambda x + (1 - \lambda)y\) is in \(S\).

Every open ball and closed ball in normed vector space is convex.

6.5.8 Theorem(Open Mapping Theorem)

Let \((B, \Vert \Vert)\) and \((B_1, \Vert \Vert_1)\) be Banach spaces and \(L : B \to B_1\) a continuous linear mapping of \(B\) onto \(B_1\). Then \(L\) is an open mapping.

6.5.9 Definition

A point \(x\) in a topological space \((X, \tau)\) is said to be an #isolated-point if \(\{x\} \in \tau\). If \(S\) is a subset of \(X\), then the set of all limit points of \(S\), denoted by \(S'\), is said to be the #derived-set of \(S\). The set \(S\) is said to be a #perfect-set if \(S' = S\); in the case that \(S = X\), the topological space \((X, \tau)\) is said to be a #perfect-space.

6.5.10 Proposition

Topological space \((X, \tau)\) is a perfect space if and only if it has not isolated points.

Compactness

Compact Spaces

7.1.1 Definition

Let \(I\) be a set and \(O_i\), \(i \in I\), a family of subset of \(X\). Let \(A\) be a subset of \(X\). Then \(O_i\), \(i \in I\), is said to be a #covering (or a #cover ) of \(A\) if \(A \subseteq \bigcup_{i \in I} O_i\). If each \(O_i\), \(i \in I\), is an open set in \((X, \tau)\), then \(O_i\), \(i \in I\) is said to be an #open-covering of \(A\) if \(A \subseteq \bigcup_{i \in I} O_i\). A finite subfamily, \(O_{i_1}, O_{i_2}, ..., O_{i_n}\), of \(O_i\), \(i \in I\), is called a #finite-covering (of \(A\)) if \(A \subseteq O_{i_1} \cup O_{i_2} \cup ... \cup O_{i_n}\).

7.1.2 Definition

A subset \(A\) of a topological space \((X, \tau)\) is said to be #compact if every open covering of \(A\) has a finite sub covering. If the compact subset \(A\) equals \(X\), then \((X, \tau)\) is said to be a #compact-space.

A finite topology is always compact, and if a discrete topology is compact, it must be finite.

7.1.3 Proposition

The closed interval \([0, 1]\) is compact.

7.1.4 Theorem[Alexander Subbasis Theorem]

A topological space \((X, \tau)\) is compact if and only if every sub basis cover of \(X\) has a finite sub cover.

The Heine-Borel Theorem

7.2.1 Proposition

Let \(f : (X, \tau) \to (Y, \tau_1)\) be a continuous surjective map. If \((X, \tau)\) is compact, then \((Y, \tau_1)\) is compact.

So compactness is a topological property.

7.2.2 Proposition

Every closed subset of a compact space is compact.

7.2.3 Proposition

A compact subset of a Hausdorff topological space is closed.

7.2.4 Theorem[Heine-Borel Theorem]

A subset of \(\mathbb{R}^n\), \(n \ge 1\), is compact if and only if it is closed and bounded.

7.2.5 Definition

A subset \(A\) of a metric space \((X, d)\) is said to be #bounded if there exists a real number \(r\) such that \(d(a_1, a_2) \le r\) for all \(a_1\) and \(a_2\) in \(A\).

7.2.6 Proposition

Let \(A\) be a compact subset of a metric space \((X, d)\). Then \(A\) is closed and bounded.

7.2.7 Proposition

Let \((X, \tau)\) be a compact space and \(f\) a continuous mapping from \((X, \tau)\) into \(\mathbb{R}\). Then the set \(f(X)\) has a greatest element and a least element.

7.2.8 Proposition

Let \(a\) and \(b\) be in \(\mathbb{R}\) and \(f\) a continuous function from \([a, b]\) into \(\mathbb{R}\). Then \(f([a, b]) = [c, d]\), for some \(c\) and \(d\) in \(\mathbb{R}\).

7.2.9 Definition

A subset \(A\) of a topological space \((X, \tau)\) is said to be #relatively-compact if its closure \(\overline{A}\) is compact.

7.2.10 Definition

A topological space \((X, \tau)\) is said to be #supercompact if there is a subbasis \(\mathcal{S}\) for the topology \(\tau\) such that if \(O_{i}, i \in I\) is any open cover of \(X\) with \(O_{i} \in \mathcal{S}\), for all \(i \in I\), then there exists \(j, k \in I\) such that \(X = O_{j} \cup O_{k}\).

7.2.11 Definition

A topological space \((X, \tau)\) is said to be #countably-compact if every countable open covering of \(X\) has a finite subcovering.

A topological space \((X, \tau)\) is said to be #locally-compact if each point \(x \in X\) has at least one neighborhood which is compact.

\(\mathbb{R}\) is locally compact but not countably compact.

7.2.12 Definition

A topological space \((X, \tau)\) is said to be #sequentially-compact if every sequence in \((X, \tau)\) has a convergent subsequence.

Every sequentially compact space is countably compact.

7.2.13 Definition

A topological space \((X, \tau)\) is said to be #pseudocompact if every continuous function \((X, \tau) \to \mathbb{R}\) is bounded.

Any pseudocompact Hausdorff normal space is countably compact.

Finite Products

The Product Topology

8.1.1 Definition

Let \((X_{1}, \tau_{1}), (X_{2}, \tau_{2}), ..., (X_{n}, \tau_{n})\) be topological spaces. Then the #product-topology \(\tau\) on the set \(X_{1} \times X_{2} \times ... \times X_{n}\) is the topology having the family \(\{O_{1} \times O_{2} \times ... \times O{n}, O_{i} \in \tau_{i}, i = 1, 2, ..., n\}\) as a basis. The set \(X_{1} \times X_{2} \times ... \times X_{n}\) with the topology \(\tau\) is said to be the #product-of-the-spaces and is denoted by \((X_{1} \times X_{2} \times ... \times X_{n}, \tau)\).

Finite-closed is not preserved via product.

8.1.2 Proposition

Let \(\mathcal{B}_{1}, \mathcal{B}_{2}, ..., \mathcal{B}_{n}\) be basis for the topological spaces \((X_{1}, \tau_{1}), (X_{2}, \tau_{2}), ..., (X_{n}, \tau_{n})\) respectively. Then the family of sets \(\{O_{1} \times O_{2} \times ... \times O_{n} : O_{i} \in \mathcal{B}_{i}, i = 1, ..., n\}\) is a basis for the product topology.

8.1.3 Proposition

Let \(C_{1}, C_{2}, ..., C_{n}\) be closed subsets of the topological spaces \((X_{1}, \tau_{1}), (X_{2}, \tau_{2}), ..., (X_{n}, \tau_{n})\) respectively. Then \(C_{1} \times C_{2} \times ... \times C_{n}\) is a closed subset of the topological space \((X_{1} \times X_{2} \times ... \times X_{n}, \tau)\).

Projections onto Factors of a Product

8.2.1 Definition

Let \(\tau_{1}\) and \(\tau_{2}\) be topologies on a set \(X\). Then \(\tau_{1}\) is said to be a #finer-topology than \(\tau_{2}\) (and \(\tau_{2}\) is said to be a #coarser-topology than \(\tau_{1}\)) if \(\tau_{1} \supseteq \tau_{2}\).

So the discrete topology is the finest topology.

8.2.2 Definition

Let \((X, \tau)\) and \((Y, \tau_{1})\) be topological spaces and \(f\) a mapping from \(X\) into \(Y\). Then \(f\) is said to be an #open-mapping if for every \(A \in \tau\), \(f(A) \in \tau_{1}\). The mapping \(f\) is said to be a #closed-mapping if for every closed set \(B\) in \((X, \tau)\), \(f(B)\) is closed in \((Y, \tau_{1})\).

8.2.3 Proposition

Let \((X_{1}, \tau_{1}), (X_{2}, \tau_{2}), ..., (X_{n}, \tau_{n})\) be topological spaces and \((X_{1} \times X_{2} \times ... \times X_{n}, \tau)\) their product space.

For each \(i \in \{1,...,n\}\), let \(p_{i} : X_{1} \times X_{2} \times ... \times X_{n} \to X_{i}\) be the projection mapping. Then

  1. each \(p_{i}\) is a continuous surjective open mapping, and
  2. \(\tau\) is the coarsest topology on the set \(X_{1} \times X_{2} \times ... \times X_{n}\) such that each \(p_{i}\) is continuous.

The projections from \(\mathbb{R}^{n}\) onto \(\mathbb{R}\) are continuous open mappings.

8.2.4 Proposition

Let \((X_{1}, \tau_{1}), (X_{2}, \tau_{2}), ..., (X_{n}, \tau_{n})\) be topological spaces and \((X_{1} \times X_{2} \times ... \times X_{n}, \tau)\) the product space. Then each \((X_{i}, \tau_{i})\) is homeomorphic to a subspace of \((X_{1} \times X_{2} \times ... \times X_{n}, \tau)\).

Tychonoff’s Theorem for Finite Products

8.3.1 Theorem[Tychonoff’s Theorem for Finite Products]

If \((X_{1}, \tau_{1}), (X_{2}, \tau_{2}), ..., (X_{n}, \tau_{n})\) are compact spaces, then \(\Pi_{i=1}^{n}(X_{i}, \tau_{i})\) is a compact spaces.

This is sufficient to give a proof of generalized Heine-Borel Theorem by transferring to \([-M, M]\).

8.3.2 Proposition

Let \((X_{1}, \tau_{1}), (X_{2}, \tau_{2}), ..., (X_{n}, \tau_{n})\) be topological spaces. If \(\Pi_{i=1}^{n}(X_{i}, \tau_{i})\) is compact, then each \((X_{i}, \tau_{i})\) is compact.

8.3.3 Definition

Let \((X, \tau_{1})\) be a topological space. Then it is said to be #locally-compact if each point \(x \in X\) hat at least one neighborhood which is compact.

Products and Connectedness

8.4.1 Definition

Let \((X, \tau)\) be a topological space and let \(x\) be any point in \(X\). The #component in \(X\) of \(x\), \(C_{X}(x)\) is defined to be the union of all connected subsets of \(X\) which contain \(x\).

8.4.2 Proposition

Let \(x\) be any point in a topological space \((X, \tau)\). Then \(C_{X}(x)\) is connected.

And \(C_{X}(x)\) is the largest connected subset of \(X\) that contains \(x\).

8.4.3 Lemma

Let \(a\) and \(b\) be points in a topological space \((X, \tau)\). If there exists a connected set \(C\) containing both \(a\) and \(b\) then \(C_{X}(a) = C_{X}(b)\).

8.4.4 Proposition

Let \((X_{1}, \tau_{1}), (X_{2}, \tau_{2}),…,(X_{n}, \tau_{n})\) be topological spaces. Then \(\Pi_{i=1}^{n}(X_{i}, \tau_{i})\) is connected if and only if each \((X_{i}, \tau_{i})\) is connected.

8.4.5 Definition

A topological space is said to be a #continuum if it is compact and connected.

Finite product of continuums is also a continuum.

8.4.6 Definition

A topological space is said to be a #compactum if it is compact and metrizable.

8.4.7 Definition

A topological space \((X, \tau)\) is said to be #locally-connected if it has a basis \(B\) consisting of connected sets.

Fundamental Theorem of Algebra

8.5.1 Theorem[The Fundamental Theorem of Algebra]

Every polynomial \(f(z) = a_{n}z^{n} + a_{n-1}z^{n-1} + … + a_{1}z + a_{0}\), where each \(a_{i}\) is a complex number, \(a_{n} \neq 0\), and \(n \ge 1\), has a root; that is, there exists a complex number \(z_{0}\) such that \(f_{z_{0}} = 0\).