Topology MOC

Topological basis

Given a topological space (𝑋,T), a basis for that space is a set of open subsets B ⊆T that can form all open subsets under union. #m/def/topology

The topology T generated by unions of subsets in B will always be the coarsest topology such that B ⊆T, a condition which also holds for Topological subbasis. In other words, B is “completed” to form a topology with the minimum additions possible, and this completion is unique. It follows from this that 𝑈 ∈T if and only if for all 𝑥 ∈𝑈 there exists 𝐵 ∈B such that 𝑥 ∈𝐵 ⊆𝑈. Any such 𝐵 for a given 𝑥 is called a basic open neighbourhood of 𝑥.

Possible bases

A given set of subsets B ⊆2𝑋 can form a basis for a topology of 𝑋 if and only if

1. For all 𝑥 ∈𝑋 there exists 𝐵 ∈B such that 𝑥 ∈𝐵.
2. For all 𝐴,𝐵 ∈B, if 𝑥 ∈𝐴 ∩𝐵, then there exists at least one 𝐶 ∈B such that 𝑥 ∈𝐶 ⊆𝐴 ∩𝐵.

The former condition comes from the requirement of a Topological space that the topology contains the whole set, and the latter comes from the requirement that any intersection of subsets in the topology is also in the topology.¹ An arbitrary collection of subsets that doesn’t meet these conditions can still be used to generate a topology, see Topological subbasis.

Examples

• For a Metric space the induced topology has open balls as its basis.

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Footnotes

1. And that our “generating operation” is just the union, so we can’t get intersections for free. ↩