Topology MOC

Topological basis

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

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

Possible bases

A given set of subsets can form a basis for a topology of if and only if

1. For all there exists such that .
2. For all , if , then there exists at least one such that .

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. ↩