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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.

Properties

If $\mathcal{T}_1, \mathcal{T}_2$ are topologies on $X$ with $\mathcal{T}_1 \subsetneq \mathcal{T}_2$, we say $\mathcal{T}_1$ is coarser than $\mathcal{T}_2$ and $\mathcal{T}_2$ is finer than $\mathcal{T}_1$. I find it helpful to remember that the indiscrete topology is the coarsest topoology, and the discrete topology is the finest.

Types

There are various additional separation axioms that give the topological space nice properties; perhaps the most interesting one is the Hausdorff condition.

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