Topology MOC

Topological space

Abstractly, a topological space consists of a set and a collection of subsets such that¹ topology

1. contains at least and .
2. Any finite or infinite union of subsets in is also in .
3. Any finite intersection of subsets in is also in .

where is called a topology on , and is said to contain open subsets of . A subset of is called closed iff its compliment is open. Thus, in any topological space the subsets and are clopen^[Simultaneously open and closed.].

On any set we can easily form the Discrete topology (every set is clopen) and the Trivial topology .

Two topologies on the same space can be compared in terms of Coarseness and fineness of topologies.

A topology can be generated by a Topological basis.

Properties

• The intersection of topologies on a fixed set is again a topology
• Coarseness and fineness of topologies

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tidy | sembr | en

Footnotes

1. 2020, Topology: A categorical approach, §0.1, p. 1 ↩