However, if we consider a locally finite collection of closed sets, the union is closed.

Every locally finite collection of subsets of a topological space X is also point-finite.

A σ-locally finite base is a base which is a union of countably many locally finite collections of open sets.

In order that a Tychonoff space X is pseudocompact it is necessary and sufficient that every locally finite collection of non-empty open sets of X is finite.

A collection of subsets of a space X is countably locally finite (or σ-locally finite) if it is the union of a countable collection of locally finite collections of subsets of X.

To see this we note that if x is a point outside the union of this locally finite collection of closed sets, we merely choose a neighbourhood V of x that intersects this collection at only finitely many of these sets.