Topological space

Coarseness and fineness of topologies

Given two topologies on the same set , , if then is said to be coarser¹ than , since it contains larger chunks of in a smaller quantity. Likewise is finer² than . Clearly all topologies are coarser than the Discrete topology and finer than the Trivial topology.³ topology

• ::: is coarser than
• ::: is finer than

Properties

• An intersection of topologies will clearly be coarser than both topologies.
• Given two topologies on the same set , if is continuous, then clearly , i.e. is finer.

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tidy | sembr | en | topology

Footnotes

1. German gröber ↩

2. German feiner ↩

3. 2020, Topology: A categorical approach, §0.1, p. 2 ↩