General Topology¶

Topological spaces¶

A topology on a set \(X\) is a set \(\tau\) of subsets of \(E\) [1] that satisfies the following conditions:

\(E\in\tau\)
If \((A_\lambda)_{\lambda\in L}\) is a family of subsets of \(X\) such that \(A_\lambda\in\tau\), \(\forall\lambda\in L\), then \(\bigcup_{\lambda\in L}A_\lambda\in\tau\). [2]
If \(A\) and \(B\) are elements of \(\tau\), then \(A\cap B\in\tau\).

A topological space is an ordered pair \((X,\tau)\) where \(X\) is a set and \(\tau\) is a topology on \(X\). Given a topological space \((X,\tau)\), by slight abuse of terminology the set \(X\) itself is also called a topological space. The elements of \(\tau\) are the open sets of the topological space \(X\).

It follows from the second property for \(L=\emptyset\) that \(\emptyset\in\tau\). That is, the empty set is an open set.

From the last property, it follows by induction that if \((A_k)_{k=1}^K\) is a finite family of elements of \(\tau\), then \(\bigcap_{k=1}^K A_k\in\tau\).

A topology on a set \(X\) is, therefore, a collection of subsets of \(X\) that contains \(X\) itself and is closed under arbitrary unions and countable intersections.

[1]	In other words, \(\mathfrak{D}\) is a subset of \(\mathfrak{P}(X)\)

[2]	The index set \(L\) is not necessarily countable and it can be empty.