# Does closed set contain only boundary points or interior points also?

I am reading this.

It says

Intuitively, an open set is a solid region minus its boundary. If we include the boundary, we get a closed set, which formally is defined as the complement of an open set.

Now, question is if a closed set includes interior points also then how can it be complement?

I know basic set theory. Enlighten me! :)

Thanks!

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The closed set you get by including the boundary is not the complement of the previously mentioned open set. It is the complement of a different open set, namely the "outside" of the solid region. –  Rahul Dec 31 '10 at 7:09
Helpful comment thanks! –  Pratik Deoghare Dec 31 '10 at 7:20

## 1 Answer

A Closed set is by definition a set whose complement is an open set. Note that this also includes the possibility that a set is both open and closed, for example in a space with two connected components, each component is both open and closed.

Now, in what you have highlighted the complement of the solid region (inclusive of boundary) i.e. the whole space without the region, is open. Which, means that the solid region (inclusive of boundary) is closed.

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How strange... when I was typing this a few seconds ago there were no replies and now there are two, posted more than 5 mins ago, is this a fault of my browser or some lag problems with the site? –  Dactyl Dec 31 '10 at 7:17
I'm surprised you didn't get an alert about my answer being posted. I deleted it anyway, partly because it seemed redundant. –  Jonas Meyer Dec 31 '10 at 7:21
I don't understand. There was some answer earlier which I think I understood but its deleted. I don't know any topology(connected components?). –  Pratik Deoghare Dec 31 '10 at 7:24
Yeah I didn't get an alert, but that is most probably because the answers were already posted while I was typing mine but the browser was not showing them. Perhaps this has something to do with chrome. –  Dactyl Dec 31 '10 at 7:30
@TheMachineCharmer: A space is called disconnected if we can find two disjoint open sets that together form the whole space. In which case the complement of one is the other, therefore, each is both open and closed. –  Dactyl Dec 31 '10 at 7:34