# Rao: Proposition 1.2.4. Superfluous section of proof?

1. Jun 19, 2012

### Rasalhague

Rao: Topology: Proposition 1.2.4. If (X,T) is a topological space, a subset A of X is closed iff the the derived set of A is a subset of A: $A'\subseteq A$.

Rao's proof of $(A'\subseteq A) \Rightarrow (X\setminus A \in T)$ goes like this:

To me, this looks like enough to show that $(A'\subseteq A) \Rightarrow (X\setminus A \in T)$, since a set A is open iff each of its points belongs to a neighborhood which is a subset of A. So $X\setminus A$ is open. In other words, A is closed.[/QUOTE]

But Rao goes on:

This seems superfluous to me. Am I missing something? Why not just say U is the neighborhood of x that's a subset of $X\setminus A$?

2. Jun 19, 2012

### micromass

Staff Emeritus
Your proof looks fine. What Rao says isn't wrong, but it can be shortened.