# Is Discreteness A Topological Property?

1. Sep 20, 2011

### strman

Is discreteness a topological property?

Hey guys. I'm currently in an advanced calculus course (not topology), and the only mention of topological property in my text is that it's a property that is conserved under continuity. This section is just a brief primer on compact sets and continuity, but it concludes with the theorem that the continuous image of a compact set is compact. While I have an idea that the answer is yes (and I've verified this online), I'm struggling to phrase why given the tools that I have. So far I know that every mapping whose domain is discrete must be continuous, so I feel like that will help, but I'm not sure where to go from there.

Again, I don't really think I have the tools for a formal proof, but I want show that I comprehend why this is true. At the moment I just have intuition and a possible direction hinting my intuition is valid, and that's not really an adequate explanation. Thanks for any help in advance.