Separable and first-countable not implies second-countable

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This article gives the statement and possibly, proof, of a non-implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property need not satisfy the second topological space property
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Verbal statement

It is possible for a topological space to be both separable and first-countable, but not second-countable.

Definitions used

Separable space

Further information: Separable space

A topological space is termed separable if it has a countable dense subset.

First-countable space

Further information: First-countable space

A topological space is termed first-countable if for every point in the space, there is a countable basis at that point.

Second-countable space

Further information: Second-countable space

A topological space is termed second-countable if it admits a countable basis.


Any second-countable space is both first-countable and separable. Further information: Second-countable implies first-countable, second-countable implies separable


Sorgenfrey line

The Sorgenfrey line, which is defined as the real numbers given the lower limit topology for the usual ordering, is first-countable and separable but not second-countable.

For full proof, refer: Sorgenfrey line is first-countable, Sorgenfrey line is separable, Sorgenfrey line is not second-countable