J. Munkres. Topology (2013) Chapter 2: Topological Spaces and Continuous Functions. 12: Topological Spaces

I'm going to speed up, otherwise we'll never get to homotopy type theory. I'm not going to write out proofs if I can do them in my head, and I'll only write solutions to exercises here if I cannot do them in my head.

Definition: Topology

A topology on a set \(X\) is a set \(\mathcal{T}\) of subsets of \(X\), whose elements are called open sets, such that:

  1. \(X \in \mathcal{T}\)
  2. The union of any subset of \(\mathcal{T}\) is in \(\mathcal{T}\).
  3. The intersection of two elements of \(\mathcal{T}\) is in \(\mathcal{T}\).

A topological space is a pair \((X, \mathcal{T})\) such that \(X\) is a set and \(\mathcal{T}\) is a topology on \(X\).

Example 2

For any set \(X\), the discrete topology on \(X\) is \(\mathcal{P} X\). The indiscrete topology or trivial topology on \(X\) is \(\{\emptyset, X\}\).

Example 3

For any set \(X\), the finite complement topology on \(X\) is \(\{U \in \mathcal{P} X \mid X - U \text{ is finite}\} \cup \{\emptyset\}\).

Definition

Let \(\mathcal{T}\) and \(\mathcal{T}'\) be topologies on the same set \(X\). We say \(\mathcal{T}'\) is finer than \(\mathcal{T}\), and \(\mathcal{T}\) is coarser than \(\mathcal{T}'\), iff \(\mathcal{T} \subseteq \mathcal{T}'\). We say \(\mathcal{T}\) and \(\mathcal{T}'\) are comparable iff \(\mathcal{T} \subseteq \mathcal{T}'\) or \(\mathcal{T}' \subseteq \mathcal{T}\).

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