Weakly contractible space

Equivalent definitions in tabular format
A nonempty topological space is said to be weakly contractible if it satisfies the following equivalent conditions. The empty space is generally excluded from consideration when considering the question of weak contractibility.

As we see below, each of the definitions (implicitly or explicitly) implies that the space is a path-connected space.

Metaproperties
Since the homotopy group of the product of two spaces is the product of their homotopy groups, the product of two weakly contractible spaces is again weakly contractible.

Any retract, and more generally, any homotopically injective subspace of a weakly contractible space is again weakly contractible.