nLab semi-locally simply-connected topological space

Idea

A topological space $X$ is (semi-)locally simply connected if every neighborhood of a point has a subneighbourhood in which loops based at the point in the subneighborhood can be contracted in $X$. It is similar to but weaker than the condition that every neighborhood of a point has a subneighborhood that is simply connected. This latter condition is called local simple-connectedness.

Definition

A topological space $X$ is semi-locally simply-connected if it has a basis of neighbourhoods $U$ such that the inclusion $\Pi_1(U) \to \Pi_1(X)$ of fundamental groupoids factors through the canonical functor $\Pi_1(U) \to codisc(U)$ to the codiscrete groupoid whose objects are the elements of $U$. The condition on $U$ is equivalent to the condition that the homomorphism $\pi_1(U, x) \to \pi_1(X, x)$ of fundamental groups induced by inclusion $U \subseteq X$ is trivial.

• A semi-locally simply connected space need not be locally simply connected. For a simple counterexample, take the cone on the Hawaiian earring space.

Examples

Example

(circle is locally simply connected)

The Euclidean circle

$$S^1 \;=\; \big\{ x \in \mathbb{R}^2 \;\big\vert\; {\Vert x\Vert} = 1 \big\} \;\subset\; \mathbb{R}^2$$

is locally simply connected

Proof

By definition of the subspace topology and the defining topological base of the Euclidean plane, a base for the topology of $S^1$ is given by the images of open intervals under the local homeomorphism

$$\big(cos(-), sin(-)\big) \;\colon\; \mathbb{R}^1 \longrightarrow S^1 \,.$$

But these open intervals are simply connected (this example).

• A semi-locally simply connected space need not be locally simply connected. For a simple counterexample, take the cone on the Hawaiian earring space.

Application

Semi-local simple connectedness is the crucial condition needed to have a good theory of covering spaces, to the effect that the topos of permutation representations of the fundamental groupoid of $X$ is equivalent to the category of covering spaces of $X$.

This is the fundamental theorem of covering spaces, see there for more.

In topos theory

For a topos-theoretic notion of locally $n$-connected see locally n-connected (infinity,1)-topos.

Related concepts

• locally path-connected topological space