Topologies

Why

We want to generalize the notion of continuity.

Definition

Given a set $X$, a topology on $X$ is a set of subsets of $X$ for which (1) the empty set base set are distinguished (2) the intersection of a finite family of distinguished subsets is distinguished, and (3) the union of a family of distinguished subsets is distinguished. We call the elements of the topology the open sets.

A topological space is an ordered pair: a base set and a set distinguished subsets of the base set which are a topology.

Notation

Let $X$ be a non-empty set. For the set of distinguished sets, we tend to use $\mathcal{T} $, a mnemonic for topology, read aloud as “script T”. We tend to denote elements of $\mathcal{T} $ by $O$, a mnemonic for open. We denote the topological space with base set $X$ and topology $\mathcal{T} $ by $(X, \mathcal{T} )$. We denote the properties satisfied by elements of $\mathcal{T} $:

1. $X, \varnothing \in \mathcal{T} $
2. if $O_1, \dots , O_n \in \mathcal{T} $, then $\bigcap_{i = 1}^{n} O_i \in \mathcal{T} $
3. if $O_\alpha \in \mathcal{T} $ for all $\alpha \in I$, then $\bigcup_{\alpha \in I} \in \mathcal{T} $

Examples

$\R $ with the open intervals as the open sets is a topological space.