PlanetMath: semilocally simply connected

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semilocally simply connected

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A topological space is semilocally simply connected if, for every point $x \in X$ , there exists a neighborhood of such that the map of fundamental groups

$\displaystyle \pi_1(U,x) \longrightarrow \pi_1(X,x)$

induced by the inclusion map $U \hookrightarrow X$ is the trivial homomorphism.

A topological space is connected, locally path connected, and semilocally simply connected if and only if it has a universal cover.

"semilocally simply connected" is owned by djao.

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Other names:

semilocally 1-connected, locally relatively simply connected

Attachments:: example of a space that is not semilocally simply connected (Example) by mathcam
example of a semilocally simply connected space which is not locally simply connected (Example) by antonio

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AMS MSC:	54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces )
	57M10 (Manifolds and cell complexes :: Low-dimensional topology :: Covering spaces)

Pending Errata and Addenda

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