Topological Spaces: What are they?

Question

Let $X\subseteq \Bbb R$, $\tau$ is a topology on $X$.

What even is $(X,\tau)$? Is it an ordered pair? Is it $\{(x,y)|x\in X, y\in \tau\}$? Is it a subset of $\Bbb R^2$? I (pretty much) know what topologies are, but I hove no idea how to think of $(X,\tau)$ visually.

Answer 1

It is (math is) all about sets and structures. With your permission, I'll expand some discussion here.

Knowing (even intuitively) what is a set, one can ask for distinct structures on that sets.

If $G$ is a set and $\star\colon G\times G\to G$ is an operation in $G$ satisfying

• $\star(\star(a,b),c) = \star(a,\star(b,c))$ for all $a,b,c\in G$;
• there exists $e\in G$ such that $\star(a,e)=\star(e,a)=a$ for all $a\in G$;
• for all $a\in G$ there exists $b\in G$ such that $\star(a,b)=\star(b,a)=e$;

then we say $\langle G, \star\rangle$ is a group. Notice that $G$ itself is just a set, while the pair $\langle G, \star\rangle$ is a group. People say "$G$ is a group" just for simplicity (when the operation is clear), though it is an abuse of notation.

Further, if we have a set $R$ with two operations $+\colon R\times R\to R$ and $\bullet\colon R\times R\to R$ satisfying some axioms, the triple $\langle R, +, \bullet\rangle$ is a ring, while $R$ itself is just a set. People say "$R$ is a ring" for simplicity (when the operations are clear), but it is also an abuse of notation.

This list could be infinite.. let us focus on your question.

A topological space is a pair $\langle X, \tau\rangle$, where $X$ is some set and $\tau\subseteq 2^X$ (the power set of $X$) satisfies

• $\emptyset, X \in \tau$;
• if $\mathcal{C}\subseteq\tau$ then $\bigcup\mathcal{C}\in \tau$;
• if $A_1,\cdots,A_n\in\tau$ then $\bigcap_{i=1}^n A_i \in\tau$.

The set $X$ is then called the base set and $\tau$ is called a topology on $X$. The elements of $\tau$ are called open subsets of $X$. People say "$X$ is a topological space" for simplicity (when the topology is clear), though it is an abuse of notation.

Answer 2

We consider $$( {X},\tau)$$ as an ordered pair in which the first component is a set and the second component is a family of subsets of $X$ called the topology on $X$. Elements of $\tau$, are called open sets of $X$ and they satisfy some axioms. As you know we like to have the empty set to be open and the intersection of finitely many open sets to be open and the union of open sets to be open and $X$ to be open. Since we can have more than one topology defined on a set $X$, we consider $$( {X},\tau)$$ to distinguish $\tau $ from other topologies on $X$

Answer 3

The essential part of an ordered pair is the following: $(a,b) = (c,d)$ iff $a=c$ and $b=d$, so two ordered pairs are the same iff both components of the pair are the same. The Kuratowski or other definitions are just implementation details, but this property is the crucial one: a pair is uniquely determined by its two components (and in order, so $(a,b) != (b,a)$ as well if $a \neq b$).

A topological space is a set with an extra structure on it, a family of subsets of $X$ satisfying some definitions that make it into a topology. Writing it as $(X,\tau)$ wants to express that we care about the combination of both, so if $\tau'$ is some different collection of subsets of $X$, automatically $(X,\tau) \neq (X, \tau')$, because the second components are different. And of course if $X \neq X'$ we also have $(X, \tau) \neq (X', \tau)$ by the essential property of a pair.

So if we are given a topological space $(X,\tau)$ the unique first component is the set on which the topology is defined and the unique second component is the set of subsets of $X$ etc. As said, it's just a handy notation to capture the "unique combination" of two things.