Register to reply

Continuity and compactness

by jjou
Tags: compactness, continuity
Share this thread:
jjou
#1
Oct8-08, 11:03 PM
P: 64
(Problem 62 from practice GRE math subject exam:) Let K be a nonempty subset of [tex]\mathbb{R}^n[/tex], n>1. Which of the following must be true?

I. If K is compact, then every continuous real-valued function defined on K is bounded.
II. If every continuous real-valued function defined on K is bounded, then K is compact.
III. If K is compact, then K is connected.

I know (I) is true and (III) is not necessarily true. I'm working on (II), which the answer key says is true, but I can't seem to prove it. I tried using several versions of the definition of compactness:

Closed and bounded-
K is obviously bounded if you take the function f(x)=x. Then f(K)=K is bounded.
To show K is closed, I assumed it wasn't: there is some sequence [tex](x_n)\subseteq K[/tex] such that the sequence converges to a point c outside of K. Then the sequence [tex]f(x_n)\subseteq f(K)[/tex] must converge to some point d, not necessarily in f(K). I'm not sure where to go from there or what contradiction I am looking for.

Covers / finite subcovers-
Let [tex]\{U_i\}[/tex] be any open cover of K. Then [tex]\{f(U_i)\}[/tex] is a cover of f(K). What I'd like to do is somehow force a finite subset of [tex]\{f(U_i)\}[/tex] to be a cover of f(K) - possibly using the fact that f(K) is bounded - and thus find a finite subcover for K. The problem is that I don't know that I can find a subcover for f(K).

Any ideas?
Phys.Org News Partner Science news on Phys.org
Bees able to spot which flowers offer best rewards before landing
Classic Lewis Carroll character inspires new ecological model
When cooperation counts: Researchers find sperm benefit from grouping together in mice
morphism
#2
Oct8-08, 11:56 PM
Sci Advisor
HW Helper
P: 2,020
The function f(x)=x isn't 'good', because it can't possibly be defined on K - K is a subset of R^n. To show that K is bounded, use the projection maps instead.

To show that K is closed, you can continue with what you're doing. Can you use the point c=(c_1,...,c_n) to define a continuous, but unbounded, real-valued function on K?
jjou
#3
Oct9-08, 12:26 AM
P: 64
Just to clarify - when they say 'real-valued,' they mean the image is strictly in [tex]\mathbb{R}[/tex], not simply in [tex]\mathbb{R}^n[/tex]?

I wanted to use the function [tex]f(x)=\frac{1}{|x-c|}[/tex] which is defined for x in K. This function would be unbounded, but is it still continuous?




-----------------------------------------------------
Please check -
To use the projection maps to show boundedness of K:

For each [tex]1\leq i\leq n[/tex], let [tex]f_i(x_1,...,x_n)=x_i[/tex]. Then define each [tex]K_i[/tex] to be such that [tex]f_i(K)\subset K_i\subset\mathbb{R}[/tex]. Then [tex]K\subset K_1\times K_2\times...\times K_n[/tex] is bounded.


Thanks! :)

morphism
#4
Oct9-08, 12:28 PM
Sci Advisor
HW Helper
P: 2,020
Continuity and compactness

Quote Quote by jjou View Post
Just to clarify - when they say 'real-valued,' they mean the image is strictly in [tex]\mathbb{R}[/tex], not simply in [tex]\mathbb{R}^n[/tex]?
Yes, real-valued means the image lives in in [itex]\mathbb{R}[/itex].

I wanted to use the function [tex]f(x)=\frac{1}{|x-c|}[/tex] which is defined for x in K. This function would be unbounded, but is it still continuous?
Yup - it's a composition of two continuous maps (x -> 1/(x-c) and x -> |x|).


To use the projection maps to show boundedness of K:

For each [tex]1\leq i\leq n[/tex], let [tex]f_i(x_1,...,x_n)=x_i[/tex]. Then define each [tex]K_i[/tex] to be such that [tex]f_i(K)\subset K_i\subset\mathbb{R}[/tex]. Then [tex]K\subset K_1\times K_2\times...\times K_n[/tex] is bounded.
Yes, that's fine. Although on second thought I'd just use the norm map instead of projections to prove that K is bounded.
jjou
#5
Oct9-08, 11:18 PM
P: 64
Got it. Thanks so much! :)


Register to reply

Related Discussions
Challenging Compactness/Continuity Problem Calculus & Beyond Homework 1
Compactness and continuity. Differential Geometry 8
Analysis - compactness and sequentially compactness Calculus & Beyond Homework 1
How can you tell if a specific topological space is compact? Calculus 5
Compactness contradiction physics Calculus 6