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In topology, closed surface is simply defined to be the surface that has no boundary as opposed to open surfaces.

This is the layman's definition of closed surface. Example is notably a sphere.

But, I am unable to conceive the sense of the definition. How can a surface has no boundary? What does this actually mean? I'm not getting the intuition. Plz help.

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    $\begingroup$ This is unanswerable without giving your definition of "surface". Also, have you looked at the definition of the topological boundary? Additionally, what makes this a physics question rather than a math question? $\endgroup$
    – ACuriousMind
    Mar 20, 2015 at 16:14
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    $\begingroup$ An infinite plane has no boundaries, but you wouldn't describe it as closed. I think your definition of a closed surface needs more work. I agree with ACuriousMind that I think this is maths rather than physics. $\endgroup$ Mar 20, 2015 at 16:18
  • $\begingroup$ @John Rennie: Actually sir, I was studying Gaussian surface. There I got it. Nevertheless, if it is irrelevent here, I'll remove it. Thanks. Moreover, even Wikipedia introduces the topic using this definition. $\endgroup$
    – user36790
    Mar 20, 2015 at 16:25
  • $\begingroup$ @user36790: I'd leave the question here for a bit as someone may answer. It hasn't attracted any downvotes ... $\endgroup$ Mar 20, 2015 at 16:31
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    $\begingroup$ The definition of a closed surface is a compact surface with no boundary. A plane has no boundary, but it's not compact. $\endgroup$
    – DanielLC
    Mar 20, 2015 at 20:03

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If you had a spherical piece of paper, any point on the paper would be surrounded by paper on two dimensions. You could cut out a little circle with that point in the center. If you had a normal sheet of paper, most of the paper would be like that, but there'd be a boundary where the points only have paper on one side and you could only cut out a semicircle. That's what "boundary" means when dealing with surfaces.

Unfortunately, the definition you're showing is incomplete. A closed surface must also be compact. My favorite definition would be really difficult to explain, but if you're not using some really weird way of measuring distance, a simpler one will suffice. It must be closed and bounded (no relation to the "closed" and "boundary" I already mentioned). "Closed" here means that any point not on the paper is completely surrounded by points not on the paper, so you can't just have a normal sheet of paper where only the edge is missing so it technically has no boundary. "Bounded" means that it doesn't go on forever in any direction, so a plane wouldn't count.

Edit:

I think it's probably good to explain why compact is a thing. If you look at an open interval from zero to one, it's bounded. It doesn't go on forever. But you can take a continuous function of it (which preserves all the sorts of structures mathematicians love) and get something that goes on forever. For example, $f(x) = 1/x$ is continuous on that interval, and maps it to the open interval $(1,\infty)$. If you use a closed interval, you can't do that. Any continuous function of $[0,1]$ will map it to a bounded set. You could say $1/0 = \infty$, and topologists frequently do that, but adding an infinity like that messes around with the structure of the real line so much that you're less making $[0,1]$ infinite than you are making the real line finite.

Compact means that you're dealing with a set in which being finite is inherent to the structure in a way that can't be changed by something as simple as a continuous function.

A closed surface is one that doesn't go on forever but also doesn't have edges. It just loops around on itself like a sphere.

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  • $\begingroup$ If you don't mind sir, could you please tell the meaning of "compact"? $\endgroup$
    – user36790
    Mar 21, 2015 at 1:02
  • $\begingroup$ I can't delete an accepted answer. $\endgroup$
    – DanielLC
    Mar 21, 2015 at 9:39
  • $\begingroup$ No need to delete sir. This is a good answer indeed. And one who wants to get a sense of closed surface without going to deep maths, this intuitive answer'll work. So, I accepted it. $\endgroup$
    – user36790
    Mar 21, 2015 at 9:45