# A topology is made up of open sets. Why?

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.

• 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. – Brian M. Scott Oct 31 '16 at 21:53
• What does the term open mean to you if it's not a set in a topology? – Mike Pierce Oct 31 '16 at 21:53
• 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. – GEdgar Oct 31 '16 at 21:54
• mathoverflow.net/questions/19152/… – littleO Oct 31 '16 at 21:58
• 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). – Mark Fischler Oct 31 '16 at 22:03