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# List of topologies

The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.

## Counter-example topologies

The following topologies are a known source of counterexamples for point-set topology.

## Topologies defined in terms of other topologies

### Natural topologies

List of natural topologies.

### Topologies of uniform convergence

This lists named topologies of uniform convergence.

### Other induced topologies

• Box topology
• Duplication of a point: Let ${\displaystyle x\in X}$ be a non-isolated point of ${\displaystyle X,}$ let ${\displaystyle d\not \in X}$ be arbitrary, and let ${\displaystyle Y=X\cup \{d\}.}$ Then ${\displaystyle \tau =\{V\subseteq Y:{\text{ either }}V{\text{ or }}(V\setminus \{d\})\cup \{x\}{\text{ is an open subset of }}X\}}$ is a topology on ${\displaystyle Y}$ and x and d have the same neighborhood filters in ${\displaystyle Y.}$ In this way, x has been duplicated.[1]