|February 24th, 2017, 06:26 PM||#1|
Joined: Nov 2015
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, 07:09 PM||#2|
Joined: Aug 2012
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
Last edited by Maschke; February 24th, 2017 at 07:18 PM.
|closed, difference, figure, geometric, set|
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