# Showing $\overline{H}=T \setminus ({T\setminus H)}^{\mathrm{o}}.$

Let $T$ be a set and $H\subset T$. I want to show that $$\overline{H}=T \setminus ({T\setminus H)}^{\mathrm{o}}.$$ I am able to understand this intuitively, from drawing a picture and generally just thinking about it, however I am unable to formulate a proof into words. I assume a standard proof would go about showing $\overline{H} \subset T \setminus ({T\setminus H)}^{\mathrm{o}}$ and the other way round, however I am unsure how the proper argument would be formulated using the concepts of neighbourhoods of points and so on. Help would be appreciated thanks.

• $T$ is not just a set, but a topological space. You could use that a point $p$ belongs to $\overline{H}$ if and only is every neighborhood of $p$ intersects $H$. I don't know which definition of $\overline{H}$ you use, but this is one of the equivalent definitions (you may need to prove that, if you use a different definition), and then using this definition should make the proof easy, as you suggest above. – Mirko May 4 '17 at 1:41
• Proposition is false if T isn't the space. For example, let T = H = (0,1) subset R. – William Elliot May 4 '17 at 3:44