|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.
|February 25th, 2017, 07:36 PM||#3|
Joined: Nov 2015
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.
Last edited by skipjack; March 28th, 2017 at 11:44 PM.
|closed, difference, figure, geometric, set|
|Thread||Thread Starter||Forum||Replies||Last Post|
|I need examples about open, closed and neither open nor closed sets||yuceelly||Topology||3||March 8th, 2016 10:45 AM|
|drawing a closed figure||mathmaniac||Art||16||May 6th, 2015 10:30 PM|
|Why is this set not closed?||E7.5||Calculus||2||February 25th, 2014 05:56 AM|
|Closed set and closed ball||Vasily||Real Analysis||3||February 4th, 2013 01:49 PM|
|Is the union of infinite disjoint closed sets closed?||03sqq||Real Analysis||4||November 13th, 2012 04:40 AM|