In every course in basic topology one learns the standard way to define a topology on a set $X$ , how to induce it from a basis, ambient space or a collection of functions into or out of the set. There are however other ways to associate a topology to a given set $X$ which may come up in practice. Here is a list of some of the once I have come across, let me know if you think there should be more elements to the list, as I am not an expert on this stuff.

Direct constructions

Let $X$ be a set, and $\mathcal{P}(X)$ be the power set of $X$ (the set of all subsets of $X$ ). The standard definition of a topology on $X$ is a subset $\mathcal{O}\subset \mathcal{P}(X)$ satisfying the following properties

$X, \emptyset \in \mathcal{O}$
$\bigcup_{i\in I}A_i \in \mathcal{O}$ for all arbitrary subset $\{A_i\}_{i\in I}\subset \mathcal{O}$ .
$A_1\cap ... \cap A_n \in \mathcal{O}$ for all finite subset $\{A_i\}_{i= 1, ..., n}\subset \mathcal{O}$ .

One calles the elements in $\mathcal{C}$ open, and say the $V\subset X$ is closed if $X\backslash V$ is open. Similar definitions can be made for the closed sets, but then we require that the collection be invariant under finite unions and arbitrary intersections.

The Kuratowski closure operator

Let $\mathcal{C}\subset \mathcal{P}(X)$ be the closed sets in a topology on $X$ . Then for every set $V\subset X$ there exists a set $\overline{V}\in C$ which is the smalles element containing $V$ . Clearly this is given by

$\overline{V} = \bigcap_{V\subset W\in \mathcal{C}} W.$

The closure of a set $V$ satisfies the following properties

$\overline{\emptyset} = \empty$
$A\subset \overline{A}$ for all $A\subset X$
$\overline{A} = A$ for all $A\in \mathcal{C}$
$\overline{A\cup B} = \overline{A}\cup\overline{B}$ for all $A, B \subset X$ .

It turns out these properties completely determines the topology $\mathcal{C}$ . Precicely, we can define a function $\mathcal{P}(X)$ satisfying the following set of axioms

$cl(\emptyset) = \emptyset$
$A\subset cl(A)$ for all $A\subset X$
$cl(cl(A)) = cl(A)$ for all $A\subset X$
$cl(A\cup B) = cl(A)\cup cl(B)$

the the collection of subset $A = cl(A)$ determines a topology of closed sets on $X$ . This is a good exercise! (if you want to cheat the proof can be found on the wiki page).

Convergence classes

A collection $\mathcal{B} = \{ ( \{y_i\}, y_\infty)| y_\infty, \{y_i\}\subset X, ~ i= 1, 2.... \}$ of tuples is called a convergence class (or set) if the following properties are satisfied

If $\{y_i\}$ is eventually constant, say $y_i = y$ for all $i\geq n$ , then $y_\infty = y$
If $\{y_i\}$ is a subsequence of $\{v_i\}$ , and $(\{v_i\}, v)\in \mathcal{B}$ , then $(\{y_i\}, v)\in \mathcal{B}$ .
If $(\{y_i\}, y_\infty)\not\in \mathcal{B}$ then there exists a subsequence $\{v_i\}\subset \{y_i\}$ such that for all subsequences $\{w_i\}\subset \{v_i\}$ we have $(\{w_i\}, y_\infty)\not\in \mathcal{B}$ .
If $\{y_{m,n}\}\subset X$ is a double sequence for which $(\{y_{n,m}\}_{n\in \N}, y_{infty, m}), (\{y_{\infty, m}\}, y_{\infty, \infty}) \in \mathcal{B}$ , then there exists strictly increasing function $f:\N\to \N$ such that $(\{y_{nf(n)}\}, y_{\infty, \infty} ) \in \mathcal{B}$ .

Given a topology on $X$ it should be clear how we can define a convergence class on the space $X$ . The point is that the there is a bijective correspondence between sequential topologies on $X$ and convergence classes on $X$ . If $X$ is not sequential we have to work with nets in stead of sequences, and tweak the definition ever so slightly, but there nothing essential changes.

To define the topology from a convergence class we simply note that the function

$cl: \mathcal{P}(A)\to \mathcal{P}(A)\quad cl(A) := \{y\in X~|~ \exists ~ \{y_i\}\subset A \text{ such that } (\{y_i\}, y)\in \mathcal{B}\}$

is a closure operator and then proceed as above.

Topology defined by…

Here are some theories that run parallel to the one of topology but still have some significant overlap –

Uniformities [1]

A uniformity or a uniform structure is a similar construction to the topological structure defined through subsets of $X\times X$ . Let’s see how it works. An entourage of the diagonal is a subset $V\subset X\times X$ such that

$\Delta \subset V$ (contains the diagonal)
$V = -V$ where $-(x, y):= (y, x)$

That is, $V$ is one axiom short of being an equivalence relation. We define addition of two entourages by

$A+B := \{(x,z) ~|~ \thereis y\in X ~ st ~ (x,y)\in A, \text{ and } (y,z)\in B \}$

and we introduce a ball of radius $V$ , denoted by $B(x, V)$ where $x\in X$ and $V\in \mathcal{U}$ and set it to be the quantity

$B(x, V) = \{y\in X ~|~ (x, y) \in V \}.$

Now, here is the main theorem, every uniformity $\mathcal{U}$ induces a topology $\mathcal{O}$ on $X$ by

$O = \{ K\subset X ~|~ \text{for every } x\in G \text{ there is a }V\in \mathcal{U} \text{ s.t. } B(x,V)\subset G\}.$

A uniformity $\mathcal{U}$ is a collection of such entourages which satisfy the following conditions

$V\subset \mathcal{U}$ and $V\subset W$ where $W$ is any other entourage, then $W\subset \mathcal{U}$
If $V_1, V_2\in \mathcal{U}$ , then $V_1\cap V_2 \in \mathcal{U}$ .
For every $V\subset \mathcal{U}$ there exists a $W\in \mathcal{U}$ such that $V = W + W$
$\bigcap_{V\in \mathcal{U}} V = \Delta$ .

This topology is automatically a $T_1$ -space, hence we clearly cannot get all topologies in this way. But this is also the only restriction, meaning all $T_1$ topologies can be induced by a uniformity (see [1] for this and many other properties of uniformities). Together with proximities, uniformities are two of the main tools for studying compactification of $G$ -spaces.

Proximities [1]

Proximities are defined similarly to uniformities, but this time on the power set $\mathalc{P}(X)$ . A subset (or relation) $\delta \subset \mathcal{P}(X)$ is called a proximity if it satisfies the following axioms

$A\delta B \quad \Leftrightarrow \quad B\delta A$
$A\delta(B\cup C)$ if and only if $A\delta B$ or $A\delta C$
$\{x\}\delta \{y\}$ if and only if $x = y$
$\emptyset \not\delta X$ (where $A\not \delta B$ means that we do not have $A\delta B$ )
If $A\not\delta B$ then there must exists $C,D\subset X$ such that $A\not\delta C$ , $A\not\delta D$ and $C\cap D = X$ .

Just like in with uniformities, a proximity induces a topology on $X$ by noting that the operator $cl(A) := \{x\in X ~|~ \{x\}\delta A\}$ defines a closure operator on $X$ . Just as with uniformities the topology is automatically $T_1$ and every $T_1$ topology can in fact be induced by a proximity in this fashion. See [1] for more properties of proximities.

In summary

In the future I may have to cover topological analogues on topological vector spaces, like barreled and bornological structures, and coarse structures which look much like the uniform structure defined above, but for now this will have to do…

References

[1] – Engelking, Ryszard. “General topology.” (1977).
[2] – Bogachev, Vladimir I., Oleg Georgievich Smolyanov, and V. I. Sobolev. Topological vector spaces and their applications. Berlin: Springer, 2017.

Equivalent ways to define topology