§ 12. Topological Spaces

Definition 12.1 : Topology / Topological Space

A topology on a set \(X\) is a collection \(\mathcal{T}\) of subsets of \(X\) having the following properties:

1. \(\varnothing\) and \(X\) are in \(\mathcal{T}\).

2. The union of the elements of any subcollection of \(\mathcal{T}\) is in \(\mathcal{T}\).

3. The intersection of the elements of any finite subcollection of \(\mathcal{T}\) is in \(\mathcal{T}\).

A set \(X\) for which a topology \(\mathcal{T}\) has been specified is called a topological space.

Definition 12.2 : Open Set

If \(X\) is a topological space with topology \(\mathcal{T}\), we say that a subset \(U\) of \(X\) is an open set of \(X\) if \(U\) belongs to the collection \(\mathcal{T}\).

Using this terminology, one can say that a topological space is a set \(X\) together with a collection of subsets of \(X\), called open sets, such that \(\varnothing\) and \(X\) are both open, and such that arbitrary unions and finite intersections of open sets are open.

Definition 12.3 : Discrete Topology / Indiscrete (Trivial) Topology

If \(X\) is any set, the collection of all subsets of \(X\) is a topology on \(X\); it is called the discrete topology.

The collection consisting of \(X\) and \(\varnothing\) only is also a topology on \(X\); we shall call it the indiscrete topology, or the trivial topology.

Definition 12.4 : Finite Complement Topology

Let \(X\) be a set; let \(\mathcal{T}_{f}\) be the collection of all subsets \(U\) of \(X\) such that \(X-U\) either is finite or is all of \(X\). Then \(\mathcal{T}_{f}\) is a topology on \(X\), called the finite complement topology.

Proof

Let's show that \(\mathcal{T}_{f}\) is indeed, a topology on \(X\).

Both \(X\) and \(\varnothing\) are in \(\mathcal{T}_{f}\), since \(X-X\) is finite and \(X-\varnothing\) is all of \(X\).
If \(\left\{U_{\alpha}\right\}\) is an indexed family of nonempty elements of \(\mathcal{T}_{f}\), to show that \(\bigcup U_{\alpha}\) is in \(\mathcal{T}_{f}\), we compute

\[ X-\bigcup U_{\alpha}=\bigcap\left(X-U_{\alpha}\right) \]

The latter set is finite because each set \(X-U_{\alpha}\) is finite. If \(U_{1}, \ldots, U_{n}\) are nonempty elements of \(\mathcal{T}_{f}\), to show that \(\bigcap U_{i}\) is in \(\mathcal{T}_{f}\), we compute

\[ X-\bigcap_{i=1}^{n} U_{i}=\bigcup_{i=1}^{n}\left(X-U_{i}\right) \]

The latter set is a finite union of finite sets and, therefore, finite.

Definition 12.5 : Finer / Coarser / Comparable

Suppose that \(\mathcal{T}\) and \(\mathcal{T}^{\prime}\) are two topologies on a given set \(X\).
If \(\mathcal{T}^{\prime} \supset \mathcal{T}\), we say that \(\mathcal{T}^{\prime}\) is finer than \(\mathcal{T}\); if \(\mathcal{T}^{\prime}\) properly contains \(\mathcal{T}\), we say that \(\mathcal{T}^{\prime}\) is strictly finer than \(\mathcal{T}\).
We also say that \(\mathcal{T}\) is coarser than \(\mathcal{T}^{\prime}\), or strictly coarser, in these two respective situations.
We say \(\mathcal{T}\) is comparable with \(\mathcal{T}^{\prime}\) if either \(\mathcal{T}^{\prime} \supset \mathcal{T}\) or \(\mathcal{T} \supset \mathcal{T}^{\prime}\).

Other terminology is sometimes used for this concept. If \(\mathcal{T}^{\prime} \supset \mathcal{T}\), some mathematicians would say that \(\mathcal{T}^{\prime}\) is larger than \(\mathcal{T}\), and \(\mathcal{T}\) is smaller than \(\mathcal{T}^{\prime}\).