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Topology

Definition

Given a set $X$, a topology $\mathcal{T}$ on $X$ is a set of subsets of $X$ called open sets, such that:

• $X, \emptyset \in \mathcal{T}$
• $U,V \in T \Rightarrow U \cap V \in \mathcal{T}$
• $U_i \in \mathcal{T}$ for all $i \in I$ where $I$ is an indexing set $\rightarrow \bigcup_{i \in I} U_i \in \mathcal{T}$

and we say $(X, \mathcal{T})$ is a topological space.

Types

TODO separation axioms

all these pages adapted with probably insufficient credit from my university's lecture notes