01-8-2020

# Topological Spaces - 1

## 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 $$(X, \tau)$$.

# Topological Spaces - 2

## 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:

• For each $$x \in S$$, there exist $$a, b$$ in $$\mathbb{R}$$, with $$a < b$$, such that $$x \in (a, b) \subseteq S$$.

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 - 1

## 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 $$A \cap U \neq \emptyset$$).

# Limit Points - 2

## 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 - 1

## 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 - 2

## 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.