hyperconnected space

Thus, a hyperconnected space cannot be Hausdorff unless it contains only a single point.

The continuous image of a hyperconnected space is hyperconnected.

In particular, any continuous function from a hyperconnected space to a Hausdorff space must be constant.

It follows that every hyperconnected space is pseudocompact.

Every open subspace of a hyperconnected space is hyperconnected.

In mathematics, a hyperconnected space is a topological space X that cannot be written as the union of two proper closed sets.

Every hyperconnected space is both connected and locally connected (though not necessarily path-connected or locally path-connected).

The (nonempty) open subsets of a hyperconnected space are "large" in the sense that each one is dense in X and any pair of them intersects.

Examples of hyperconnected spaces include the cofinite topology on any infinite space and the Zariski topology on an algebraic variety.