nLab semi-locally simply connected topological space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



A topological space XX 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 XX. 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.


A topological space XX is semi-locally simply-connected if it has a basis of neighbourhoods UU such that the inclusion Π 1(U)Π 1(X)\Pi_1(U) \to \Pi_1(X) of fundamental groupoids factors through the canonical functor Π 1(U)codisc(U)\Pi_1(U) \to codisc(U) to the codiscrete groupoid whose objects are the elements of UU. The condition on UU is equivalent to the condition that the homomorphism π 1(U,x)π 1(X,x)\pi_1(U, x) \to \pi_1(X, x) of fundamental groups induced by inclusion UXU \subseteq X is trivial.



(circle is locally simply connected)

The Euclidean circle

S 1={x 2|x=1} 2 S^1 = \left\{ x \in \mathbb{R}^2 \;\vert\; {\Vert x\Vert} = 1\right\} \subset \mathbb{R}^2

is locally simply connected


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

(cos(),sin()): 1S 1. (cos(-), sin(-)) \;\colon\; \mathbb{R}^1 \to 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.


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 XX is equivalent to the category of covering spaces of XX.

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


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

Last revised on October 19, 2023 at 21:55:43. See the history of this page for a list of all contributions to it.