Topology/Topological Spaces

In this section, we will define what a topology is and give some examples and basic constructions.

Motivation

In Abstract Algebra, a field generalizes the concept of operations on the real number line. This general definition allows concepts about quite different mathematical objects to be grasped intuitively by comparison with the real numbers. Likewise, the concept of a topological space is concerned with generalizing the structure of sets in Euclidean spaces. Of course, for many topological spaces the similarities are remote, but nevertheless aid in judgment and guide proofs. Interesting differences in the structure of sets in Euclidean space, which have analogies in topological spaces, are connectedness, compactness, dimensionality, and the presence of "holes".

If we begin with an arbitrary set, it may not be immediately obvious what is needed to imbue it with an interesting structure. One possibility might be to define a metric on the set, but as it turns out, requiring a metric is overly restrictive. In fact, there are many equivalent ways to define what we will call a topological space just by defining families of subsets of a given set. The properties of the topological space depend on the number of subsets and the ways in which these sets overlap. Topological spaces can be fine or coarse, connected or disconnected, have few or many dimensions.

The most popular way to define a topological space is in terms of open sets, analogous to those of Euclidean Space. (In Euclidean space, an open set is intuitively seen as a set that does not contain its "boundary").

Definition of a topological space

Given a set $X$, a topology $\mathcal{T}$ on $X$ is a collection of subsets of $X$ (called open sets) with the following properties:

• The empty set and $X$ are both in $\mathcal{T}$.
• The union of any collection of open sets is an open set. That is, $S_1, S_2 \subseteq \mathcal{T} \implies S_1 \cup S_2 \in \mathcal{T}$.
• The intersection of any finite collection of open sets is an open set. That is, $A, B \in \mathcal{T} \implies A \cap B \in \mathcal{T}$.

The pair $(X,\mathcal{T})$ is called a topological space. If the topology is evident or does not need an explicit name (since we can just refer to sets in the topology as open sets), then we just say that $X$ is a topological space.

Examples of topological spaces

For any set $X$, there are two topologies we can always define on $X$:

• The Discrete topology - the topology consisting of all subsets of a set $X$.
• The Indiscrete topology (also known as the trivial topology) - the topology consisting of just $X$ and the empty set, $\emptyset$.

Metric Topology

Given a metric space $\ (X, d)\$, its metric topology is the topology induced by using the set of all open balls as the base. One can also define the topology induced by the metric, as the set of all open subsets defined by the metric. We denote the topology $\mathcal{T}$ induced from the metric d with $\mathcal{T} = top(d)$

This forms a topological space from a metric space.

If for a topological space $(X, \mathcal{T})$, we can find a metric $d$, such that $\mathcal{T} = top(d)$, then the topological space is called metrizable.

The usual topology on the real numbers

We can define a topology $\mathcal{U}$ on $\mathbb{R}$ by defining $U\subseteq\mathbb{R}$ to be in $\mathcal{U}$ if for every point $x\in U$, there is an $\varepsilon>0$ such that $(x-\varepsilon,x+\varepsilon)\subseteq U$. We call this topology the standard topology, or usual topology on $\mathbb{R}$.

The cofinite topology on any set

Let $X$ be a non-empty set. Define $\mathcal{T}$ to be the collection of all subsets $G$ of $X$ satisfying the following:

1. Either $G = \emptyset$
2. Or $X \setminus G$ is finite.

Then $\mathcal{T}$ is a topology on $X$ called the cofinite topology (or "finite complement topology") on $X$. Further, this topology turns out to be discrete if and only if $X$ is finite.

The cocountable topology on any set

Let $X$ be a non-empty set. Define $\mathcal{T}$ to be the collection of all subsets $G$ of $X$ satisfying the following:

1. Either $G = \emptyset$
2. Or $X \setminus G$ is countable.

Then $\mathcal{T}$ is a topology on $X$ called the cocountable topology (or "countable complement topology") on $X$. Further, this topology turns out to be discrete if and only if $X$ is countable.

Sets in topological spaces

Let $X$ be a topological space. There are many types of sets we can define on $X.$

• The complement of a set A in X, denoted by $A^C$, is $A^C = X \setminus A$ (that is, the entire space except for A).
• A subset $C$ is called closed if the set $C^C$ is open. Notice that the intersection of any non-zero number of closed sets is closed and the union of finitely many closed sets is closed.
• Note also that a set can be both closed and open. The trivial examples are the empty set $\emptyset$ and the entire set $X$, each of which is both closed and open. By definition, $\emptyset$ is open, so its complement, $X$, is closed. But $X$, by definition, is an open set, so $X$ is both open and closed.
• A set $N$ is called a neighborhood of a point $x \in X$, if there is an open set $U$ such that $x\in U\subseteq N.$

We now investigate some commonly occurring sets in the study of Topology.

Definition

In a topological space, a $G_{\delta }$ set is a countable intersection of open sets. A $F_{\sigma }$ set is a countable union of closed sets.

Theorem

The complement of a $F_{\sigma }$ set is $G_{\delta }$, and vice versa.

Proof:
Let A be a $F_{\sigma }$ set and let $n\in \mathbb{N}$. Then A is a countable union of closed sets, $\bigcup\limits_{n}^{{}}{A_{n}}$ such that $A_{n}$ is closed for all n. Then $A^{c}=\bigcap\limits_{n}^{{}}{\left( A_{n} \right)^{c}}$. Since $A_{n}$ is closed, $\left( A_{n} \right)^{c}$ is open, so we have a countable intersection of open sets. Hence $A^{c}$ is $G_{\delta }$.

The entirely similar proof of the other implication is left to the reader.

Theorem

In any metric space, a closed set is a $G_{\delta }$ set.
Proof:
Let X be a metric space and let $A\subseteq (X,d)$.
Define $O_{n}=\bigcup\limits_{n}{\left\{ \beta_{1/n}(x)~\left| ~x\in A \right. \right\}}$. Observe that $O_{n}$ is open for any n, and hence the union is open. Now our goal is to show that $\bar{A}=\bigcap\limits_{n}{O_{n}}$ to show that a closed set is the intersection of countably many open sets.

$\subseteq$:
Let $x\in \bar{A}$. Then $\beta_{1/n}(x)$ intersects A at some $x_{0}$ which implies $x\in \beta _{1/n}(x_{0})\subseteq O_{n}.$. This is true for any n so $x\in \bigcap\limits_{n}{O_{n}}$.

$\supseteq$:
Let $x\in \bigcap\limits_{n}{O_{n}}$ and $\varepsilon >0$. Then $\exists ~n\in \mathbb{N}$ such that $1/n<\varepsilon$. So $x\in O_{n}\Rightarrow \exists ~x_{0}$ in A such that $x\in \beta _{1/n}(x_{0})$, which implies $x\in \beta _{\varepsilon }(x_{0})$. Thus $x\in \bar{A}$.

Therefore $\bar{A}=\bigcap\limits_{n}{O_{n}}$ and is therefore a $G_{\delta }$ set.

Theorem

In usual $\mathbb{R}$, $\mathbb{Q}$ is a $F_{\sigma }$ set.

Proof:
Since $\mathbb{Q}$ with the usual topology is a metric space, every singleton such that $x\in \mathbb{Q}$ is closed. Thus, we have a countable union of closed sets, and hence $\mathbb{Q}$ is a $F_{\sigma }$ set.

Exercises

1. Prove the following are topologies:
   • The discrete topology on any set.
   • The indiscrete topology on any set.
   • The cofinite topology on any set.
   • The cocountable topology on any set.
2. Show that the cofinite (respectively, cocountable) topology on a set $X$ equals the discrete topology if and only if $X$ is finite (respectively, countable).
3. Prove that a set is open if and only if for every element within the set, there is a neighborhood contained within the set.
   
4. Show that the discrete topology is the topology induced by the discrete metric. (This is also a splendid way of remembering which is the discrete and the indiscrete topology)