Path-connected-weakly open subset

Definition
A subset of a topological space is termed path-connected-weakly open if it satisfies the following equivalent conditions:


 * Given any continuous map from a path-connected space to the given topological space, the inverse image of the subset is open
 * The subset is a union of some of the path-components of some open subset containing it

Significance
We can replace open subsets with path-connected-weakly open subsets, for homology computation tools like Mayer-Vietoris homology sequence and excision.

Stronger properties

 * Connected-weakly open subset
 * Open subset