Locally contractible space

Symbol-free definition
A topological space $$X$$ is said to be locally contractible if it satisfies the following equivalent conditions:


 * 1) It has a basis of open subsets each of which is a contractible space under the subspace topology.
 * 2) For every $$x \in X$$ and every open subset $$V \ni x$$ of $$X$$, there exists an open subset $$U \ni x $$ such that $$U \subseteq V$$ and $$U$$ is a contractible space in the subspace topology from $$X$$.

Formalisms
Note that the locally operator here means the existence of a basis of contractible spaces. It is a stronger condition than merely saying that every point is contained in a contractible open subset; rather, we are claiming that there are arbitrarily small contractible open subsets. The mere condition that every point is contained in a contractible open subset is much weaker.

Incomparable properties

 * Contractible space: A contractible space need not be locally contractible; in fact, it need not even be locally connected! An example of a contractible space that is not locally connected is the comb space. Conversely, a locally contractible space need not be contractible. For instance, any manifold is locally contractible, but manifolds such as the circle are not contractible.