Topology is a ‘subfield’ of mathematics which attempts to capture and study the minimal structure of space. General topology seeks to do this through axiomatizing the idea of neighborhoods of a point. The first definition of a topological space seems to be due to Hausdorff^{1} in 1914. We will take a short detour and first introduce metric spaces and use it to explain the intuition behind the general axioms for a topological space.
The concept of a metric space is actually a bit older than a general topological space, albeit only by approximately 8 years. A metric space is the following: $(X,d)$, where $X$ is the set of points of our space, and $d: X \times X \rightarrow \mathbb{R}$ is a metric. A metric intuitively ‘measures’ the distance between two points, and must satisfy the following axioms:
Non-negativity: $d(x,y) \geq 0$, so that the ‘distance’ is never negative.
Coincidence axiom: $d(x,y) = 0$ if and only if $x=y$, so that(with 1.) different points always have a positive distance between them.
The triangle inequality: If $x,y$ and $z$ are three points of $X$(they may be the same points), we should have that $d(x,y) + d(y,z) \geq d(x,z)$
Symmetry: $d(x,y) = d(y,x)$ for every point $x$ and $y$ of $X$.
So what do we want a neighborhood $U$ of a point $x$ to be? Well, intuitively it should contain the ‘neighbors’ of $x$. That is there should be a positive real number $r$ where for any $y \in X$, the condition $d(x,y) \lt r$ implies that $y \in U$. Let’s introduce a piece of notation to facilitate this. Let $B(x,r)$ be the ‘open ball’ of radius $r$ based at $x$, that is $B(x,r) = \{ y \in X : d(x,y) \lt r \}$. Then our intuition says we should define a ‘neighborhood’ of $x$ to be a set $U$ containing $x$, where there is a positive $r$ with $B(x,r) \subset U$.
Now we define an open subset of $X$ to be a set $U$ where $U$ is a neighborhood of every point in $U$, that is if $u$ is a point of $X$ with $u \in U$, then we can find a $r_u \gt 0$ with $B(u,r_u) \subseteq U$. Notice that the family of open subsets of $X$ has a few nice properties:
$X$ is an open subset of $X$, by taking any positive $r_x$ for each $x$ we see $X$ is a neighborhood of every point in $X$.
$\emptyset$ is an open subset of $X$, by the triviality that it has no points, so it is a neighborhood of every point inside of it.
If $U$ and $V$ are two open subsets of $X$, then $U \cap V$ is also an open subset of $X$. Let $w \in U \cap V$, so that $w \in U$ and $w \in V$. By the former, we get a $r_1 \gt 0$ with $B(w,r_1) \subseteq U$ and the latter gives $r_2 \gt 0$ with $B(w,r_2) \subseteq V$. We see that $B(w, \textrm{min}\{ r_1,r_2\}) \subseteq B(w,r_1)$ and $B(w, \textrm{min}\{ r_1,r_2\}) \subseteq B(w,r_2)$, so that $B(w, \textrm{min}\{ r_1,r_2\}) \subseteq U$ and $B(w, \textrm{min}\{ r_1,r_2\}) \subseteq V$, but then $B(w, \textrm{min}\{ r_1,r_2\}) \subseteq U \cap V$. So $U \cap V$ is a neighborhood of every point inside of it, thus an open subset of $X$.
If $\{ U_i\}_{i \in I}$ is an arbitrary collection of open sets, then the union of them $\cup U_i$ is also open. The proof is left as an exercise to the reader.
It is these 4 properties which we take as the basic properties of open sets.
A topological space is a pair $(X, \tau)$^{2}, where $\tau$ is a collection of subsets of $X$ satisfying the following properties:
$X \in \tau$ and $\emptyset \in \tau$
If $\{ U_i\}_{i \in I} \subset \tau$, then $\cup U_i \in \tau$
If $U,V \in \tau$ then $U \cap V \in \tau$
The elements of $\tau$ are called open subsets of $X$, or if the context is clear they are often merely called open sets. A set is called closed if it’s complement is open.
An non-empty set $S$ with two or more elements always has at least two topologies: the trivial topology $(S,\tau)$ and the discrete topology $(S,\tau')$. The former is $\tau = \{ \emptyset, S\}$ and the latter has $\tau' = \mathcal{P}(S)$ the power set of $S$. If $S$ has one element or no elements, these are the same.
There is almost nothing true one may say about a topological space without requiring additional properties!
Hausdorff’s axiomatization is not the modern one, it seems to include the Hausdorff condition as an axiom. ↩
You will probably never see the notation $(X,\tau)$ for a topological space again while reading and working, but a bit of completeness at this point is nice and can clarify certain discussions(like comparing different topologies on the same space). ↩