In the definition of a topological space, a topological space $(X,\tau)$ is a a pair where $X$ is the set on which the topology is defined and $\tau$ is a topology which is a subset of $\mathcal{P}(X)$ defined with specific conditions.
What I don't understand is what exactly is meant by the pair $(X,\tau)$.
What does an element from this "topological space" look like? Is it also a pair?
Is this supposed to be a tuple, since that is what the $(\cdot,\cdot)$ suggests?
Is there a way I can understand this geometrically, e.g. what would $(\mathbb{R},\tau)$ look like?
I want to understand how I can envision a topological space.
What does a *pair* mean in the definition of a topological space
general-topology
Best Answer
Pair is a pair, two things in correct order. That's it. Seriously. It's like bicycle is a pair of wheels together with some skeleton and other mechanisms. Of course in mathematics pair has concrete definition, but it is not really relevant. What matters is that pair is an ordered collection of two things.
So a topological space is a set together with some additional structure, called topology. The reason we say "a topological space is a pair $(X,\tau)$ such that..." is because the topological space depends on both the underlying set $X$ and the topology $\tau$. I.e. on a given set $X$ we can define multiple different topological structures, e.g. $\tau=\{\emptyset, X\}$ (a.k.a. antidiscrete topology) and $\tau=P(X)$ (a.k.a. discrete topology) are two different (unless $|X|\leq 1$) topologies on $X$.
Strictly, formally speaking, we never talk about elements of a topological space (since formally any topological space $(X,\tau)$ has exactly two elements: $X$ and $\tau$). When we say that $x$ is an element of a topological space $(X,\tau)$ then typically what we mean is that $x\in X$, i.e. we only talk about elements of the underlying set.
That depends on $\tau$. When $\tau$ is the Euclidean topology, then we deal with the standard Euclidean line. But there are infinitely many nonequivalent topologies $\tau$ on $\mathbb{R}$. Some of them easy to understand, e.g. discrete topology on $\mathbb{R}$ makes every point isolated. But some of them are quite abstract and weird to wrap mind around, e.g. high dimensional spaces (note that since $\mathbb{R}$ is equinumerous with any $\mathbb{R}^n$ then for any $n$ we can define topology on $\mathbb{R}$ making it $n$-th dimensional).