About the concept of locally finite topological space

Question

I have been defined "locally finite space" as follows:

Let $X$ be a topological space. We say that it's locally finite if for any $x\in X$ we have a finite neighbourhood containing it.

What does it mean "finite neighbourhood"? Is that we have a finite number of open sets covering that neighbourhood? Or it means that it contains a finite number of points of $X$?

Thanks for your time.

Answer 1

It means there exists a neighborhood of $x$ which is finite. An example would be a set endowed with the discrete topology: any $x$ admits $\{x\}$ as a finite neighborhood.

Answer 2

A topology is locally finite if every point is contained in a finite open set - that is an open set with a finite number of points.

Obviously all finite topologies are locally finite but here are plenty of examples of infinite but locally finite topologies. For instance if $X$ is an infinite set then let $D$ be the set of cofinite subsets of $X$, then, thinking of $D$ as a poset under set inclusion, the Alexandroff topology on $D$ is locally finite. If you want more note that subspaces and finite products of locally finite spaces are locally finite.