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-   -   Closed set (metric spaces) (http://www.physicsforums.com/showthread.php?t=548254)

Ted123 Nov7-11 02:39 PM

Closed set (metric spaces)
 
Suppose [itex]f:\mathbb{R}\to \mathbb{R}[/itex] is a continuous function (standard metric).

Show that its graph [itex]\{ (x,f(x)) : x \in \mathbb{R} \}[/itex] is a closed subset of [itex]\mathbb{R}^2[/itex] (Euclidean metric).

How to show this is closed?

lanedance Nov7-11 03:01 PM

Re: Closed set (metric spaces)
 
what are your definitions of closed?

thinking geometrically, a continuous function will have a graph that is an unbroken curve in the 2D plane, how would you show this is closed in R^2

Ted123 Nov7-11 03:14 PM

Re: Closed set (metric spaces)
 
Quote:

Quote by lanedance (Post 3603566)
what are your definitions of closed?

thinking geometrically, a continuous function will have a graph that is an unbroken curve in the 2D plane, how would you show this is closed in R^2

Well a set [itex]A[/itex] is closed if [itex]\partial A \subset A[/itex], i.e. [itex]\partial A \cap A^c = \emptyset[/itex]

Ted123 Nov7-11 03:17 PM

Re: Closed set (metric spaces)
 
How could I show it is closed by considering the function [itex]f : \mathbb{R}^2 \to \mathbb{R}[/itex] defined by [itex]f(x,y)=f(x)- y[/itex]?


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