In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a locally simply connected space is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
that admits a
basis of
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
sets.
Every locally simply connected space is also
locally path-connected and
locally connected.

The
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
is an example of a locally simply connected space which is not simply connected. The
Hawaiian earring is a space which is neither locally simply connected nor simply connected. The
cone on the Hawaiian earring is
contractible and therefore simply connected, but still not locally simply connected.
All
topological manifolds and
CW complex
In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
es are locally simply connected. In fact, these satisfy the much stronger property of being
locally contractible.
A strictly weaker condition is that of being
semi-locally simply connected
In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space ''X'' is semi-locally simply connected i ...
. Both locally simply connected spaces and simply connected spaces are semi-locally simply connected, but neither converse holds.
References
Properties of topological spaces
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