mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a locally simply connected space is a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...

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 between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...

sets. Every locally simply connected space is also

locally path-connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness ...

and

locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness ...

The

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

is an example of a locally simply connected space which is not simply connected. The

Hawaiian earring In mathematics, the Hawaiian earring \mathbb is the topological space defined by the union of circles in the Euclidean plane \R^2 with center \left(\tfrac,0\right) and radius \tfrac for n = 1, 2, 3, \ldots endowed with the subspace topology: ...

is a space which is neither locally simply connected nor simply connected. The

cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...

on the Hawaiian earring is

contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...

and therefore simply connected, but still not locally simply connected. All

topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...

s and

CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...

es are locally simply connected. In fact, these satisfy the much stronger property of being

locally contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within tha ...

. A strictly weaker condition is that of being semi-locally simply connected. Both locally simply connected spaces and simply connected spaces are semi-locally simply connected, but neither converse holds.

References

Properties of topological spaces {{topology-stub