Examples
Any “nice” space such as a manifold or CW complex is semi-locally simply connected. In some sense, only a pathological space can fail to satisfy this condition.
A simple example of a space that is not semi-locally simply connected is the Hawaiian earring: the union of the circles in the Euclidean plane with centers (1/n, 0) and radii 1/n, for n a natural number. Give this space the subspace topology. Then all neighborhoods of the origin contain circles that are not nullhomotopic.
The Hawaiian earring can also be used to construct a semi-locally simply connected space that is not locally simply connected. In particular, the cone on the Hawaiian earring is contractible and therefore semi-locally simply connected, but it is clearly not locally simply connected.
Another example of a non-semi-locally simply connected space is the complement of Q × Q in the Euclidean plane R2, where Q denotes the set of rational numbers. In fact, the fundamental group of this space is uncountable (Hatcher p. 54).
Read more about this topic: Semi-locally Simply Connected
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