Topological Spaces

Entry conditions

Use topology only when you can specify a collection of “open sets” with closure properties.

Definitions

A topology on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ such that:

1. $\emptyset$ and $X$ are in $\mathcal{T}$.
2. Any union of sets in $\mathcal{T}$ is in $\mathcal{T}$.
3. Any finite intersection of sets in $\mathcal{T}$ is in $\mathcal{T}$.

The pair $(X,\mathcal{T})$ is a topological space.

Vocabulary (plain language)

• Open set: a subset declared open by the topology rules.
• Union: combine sets by taking anything in any of them.
• Intersection: combine sets by taking only what is in all of them.

Symbols used

• $X$: the underlying set.
• $\mathcal{T}$: the collection of open sets.

Intuition

Topology is about which subsets count as “open” so that we can talk about continuity and gluing without relying on distance.

Worked example

On any set $X$, the discrete topology declares every subset open. The indiscrete topology declares only $\emptyset$ and $X$ open.

How to recognize the structure

• You can list or describe the open sets.
• The axioms are satisfied.

Common mistakes

• Declaring opens without checking closure under unions and finite intersections.