Semilocally simply connected space

Builds on Topological space Fundamental group

Definition

Let $X$ be a topological space. The space $X$ is semilocally simply connected if every point $x \in X$ has an open neighbourhood $U$ with the following property. The homomorphism $\pi_1(U, x) \to \pi_1(X, x)$ of fundamental groups induced by the inclusion $U \hookrightarrow X$ is trivial. Equivalently, every loop in $U$ based at $x$ is null-homotopic in $X$ .

Remarks

Example: a manifold is semilocally simply connected. Each point has a neighbourhood $U$ homeomorphic to an open ball, so $\pi_1(U, x)$ is already trivial.

Non-example: the Hawaiian earring $H = \bigcup_{n \ge 1} C_n$ , where $C_n \subset \R^2$ is the circle of radius $1/n$ centred at $(1/n, 0)$ , is not. Every neighbourhood $U$ of the origin contains some $C_n$ , and a loop around $C_n$ is not null-homotopic in $H$ , because $H$ retracts onto $C_n$ .

The cone on the Hawaiian earring is contractible, so it is semilocally simply connected, but it is not locally simply connected. The spelling "semi-locally simply connected" is also common.

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Fundamental group concept
Topological space concept
Semilocally simply connected space definition

Tag	`TMJ0SFW7M7`
URI	`m2://TMJ0SFW7M7`
Kind	definition
Writing	W0 stub
MSC	57M10, 55Q05
Defines	Semilocally simply connected space
Uses	Topological space, Fundamental group
Changed	2 November 2026

Semilocally simply connected space

Definition

Remarks

Reading path

Further reading

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