My Math Forum Difference between Closed geometric figure & closed set

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 February 24th, 2017, 05:26 PM #1 Senior Member   Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2 Difference between Closed geometric figure & closed set what could be the difference between a closed set and a closed geometric figure ? Can anyone explain me with an example please.
 February 24th, 2017, 06:09 PM #2 Senior Member   Joined: Aug 2012 Posts: 2,326 Thanks: 717 Two different meanings of closed. A closed geometric figure is topologically a loop. Like a circle or a polygon. If they don't intersect themselves they're called simple closed curves. A closed set is a set that contains its limit points. The closed unit interval on the x-axis in the x-y plane is a closed set but not a closed curve or a closed geometric figure. I'm trying to think of an example the other way but drawing a blank at the moment. edit -- I think a closed curve must be a closed set. A closed curve by definition is a continuous (or maybe smooth) function from the closed unit interval to $\mathbb R^n$ (or some topological space) such that $f(0) = f(1)$. The continuous image of a compact set is compact so I think we have a proof. To sum up: The image of a closed curve must be a closed set. A closed set need not be a closed curve. edit2 -- My proof requires that the topological space be Hausdorff. Better to leave the target as Euclidean space, then my proof works. The continuous image of a compact interval is a closed set and is also by definition a curve. Fine points of the corner case here ... A compact set need not be a closed set. They didn't tell us that in real analysis http://math.stackexchange.com/questi...ets-are-closed Thanks from Joppy Last edited by Maschke; February 24th, 2017 at 06:18 PM.
February 25th, 2017, 06:36 PM   #3
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Joined: Nov 2015

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Quote:
 Originally Posted by Maschke Two different meanings of closed. A closed geometric figure is topologically a loop. Like a circle or a polygon. If they don't intersect themselves they're called simple closed curves. A closed set is a set that contains its limit points. The closed unit interval on the x-axis in the x-y plane is a closed set but not a closed curve or a closed geometric figure. I'm trying to think of an example the other way but drawing a blank at the moment. edit -- I think a closed curve must be a closed set. A closed curve by definition is a continuous (or maybe smooth) function from the closed unit interval to $\mathbb R^n$ (or some topological space) such that $f(0) = f(1)$. The continuous image of a compact set is compact so I think we have a proof. To sum up: The image of a closed curve must be a closed set. A closed set need not be a closed curve. edit2 -- My proof requires that the topological space be Hausdorff. Better to leave the target as Euclidean space, then my proof works. The continuous image of a compact interval is a closed set and is also by definition a curve. Fine points of the corner case here ... A compact set need not be a closed set. They didn't tell us that in real analysis general topology - Compact sets are closed? - Mathematics Stack Exchange

Is compact set and closed curve are same ?
Why do you take Hausdroff space ? I think all the subsets of the H. space are disjoint which implies they are open?!
I have seen that a topological space which is closed isn't a H.space then how is it possible to show it as a closed set ?

I'm new to topology, I'm learning it for my M.Sc. entrance. Kindly correct me if I'm wrong in the concept.

Thank you

Last edited by skipjack; March 28th, 2017 at 11:44 PM.

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### difference between closed and simple closed figure

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