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I've been introduced today to the notion of Topological space. In my notes, it says that a Topology on a set $X$ have elements called "open sets".

It means that a Topology cannot contain closed sets? All the subsets of $X$ in a topology must be open?

I want to understand the motivation behind the definition.

Thanks for your time!

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    $\begingroup$ Open set is simply another name for member of the topology: they mean exactly the same thing. However, some open sets are also closed sets, so a topology can contain closed sets. $\endgroup$ – Brian M. Scott Oct 31 '16 at 21:53
  • $\begingroup$ What does the term open mean to you if it's not a set in a topology? $\endgroup$ – Mike Pierce Oct 31 '16 at 21:53
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    $\begingroup$ If you know the open sets, then you know the closed sets. And vice versa. So to define a topology you can (and sometimes it is best to) specify what are the closed sets. Other ways may be to specify the convergent filters. Or the neighborhoods of each point. $\endgroup$ – GEdgar Oct 31 '16 at 21:54
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    $\begingroup$ mathoverflow.net/questions/19152/… $\endgroup$ – littleO Oct 31 '16 at 21:58
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    $\begingroup$ Something you said indicates a mis-understanding: it is not the case that all subsets of the space X must be open. The topoopgy $T$ is defined by saying which subsets of $X$ are deemed "open". So yes, all the sets in the definition of the topology are open, but not all the sets in the space (except if the topology is a pretty trivial one). $\endgroup$ – Mark Fischler Oct 31 '16 at 22:03
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You're misunderstanding what "a topology consists of open sets" means.

The point is that the open sets characterize the topology. That is, to tell you a topology, I just need to tell you the open sets - then you have all the information about it there is. For instance, a "closed" set is just the complement of an open set.

In natural language, we sometimes use the word "topology" to mean all the data provided by the topology: which sets are open, which are closed, which are neither, which are compact, etc. But this is an abuse of terminology: the topology itself really just is the "notion of openness," and everything else is derived from that.

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"Open" is definitely an unfortunate choice of name for the sets in a topology. I don't know the history of it, but I'd guess it comes from the fact that open intervals are open sets in the real line.

The reason it's unfortunate is that sets in topological spaces are not like doors. If a set is open, that doesn't prevent it from also being closed, and most sets you encounter will be neither open nor closed.

It's best to think of an open set as just being an element of a topology (that is, a topology on a space is a collection of subsets of the space, and these subsets are dubbed "open"). This is the definition that matters.

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