Definitions of locally contractible spaces

Asked Jan 29 '21 at 20:11

Active Dec 25 '24 at 01:06

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I have seen two definitions of locally contractible spaces:

(Wikipedia) A topological space $X$ is locally contractible if every point has arbitrarily small contractible neighborhoods.
(nLab) A topological space $X$ is locally contractible if every point has arbitrarily small open contractible neighborhoods.

Clearly, Definition 2 implies Definitions 1. But I don't know how to prove the converse.

Q: Are these two definitions equivalent?

asked Jan 29 '21 at 20:11

Zephyr

In many books neighborhoods are defined as open sets anyway. – Mark Jan 29 '21 at 20:16
@Mark Yes, but here a neighborhood is not assumed to be open. – Zephyr Jan 29 '21 at 20:53
Another definition is stated here, That's the one recall being taught too. – Henno Brandsma Jan 29 '21 at 23:37

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