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Let $\mathbb{X}$ be a compact metric space. $f$ is a continuous function $\mathbb{X}\to\mathbb{X}$. Which of the following necessarily be true and why?

  1. $f$ has fixed point
  2. $f$ is uniformly continuous
  3. $f$ is a closed map

And are there any tricks to solve the problem?

Thank you!

Edit: I have deleted the former duplicate question.

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You have posted this exact question before. –  Michael Greinecker Oct 29 '12 at 8:21
    
@MichaelGreinecker I'm sorry. I've voted to close. –  J.A.F Oct 29 '12 at 8:25
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Why have you deleted your original question? –  Chris Eagle Oct 29 '12 at 17:48
    
@ChrisEagle I forgot I have asked the question before.So I deleted the former duplicate question. –  J.A.F Oct 30 '12 at 17:15
    
I've noticed that you have asked quite a lot of questions recently. I wanted to make sure that you are aware of the quotas 50 questions/30 days and 6 questions/24 hours, so that you can plan posting your questions accordingly. (If you try to post more questions, stackexchange software will not allow you to do so.) For more details see meta. –  Martin Sleziak Nov 9 '12 at 7:03

2 Answers 2

up vote 5 down vote accepted

HINTS:

(1) Let $\Bbb X=\{0,1\}$ with the discrete topology, and let $f:\Bbb X\to\Bbb X:x\mapsto 1-x$. (Yes, this is a metric space; $d(0,0)=d(1,1)=0$ and $d(0,1)=d(1,0)=1$ is a metric.)

(2) It’s true; try to prove it. You may find this useful.

(3) A closed subset of a compact space is compact. What do you know about the continuous image of a compact set?

There aren’t really any tricks. (2) and (3) are standard results or immediate consequences of standard results, so with a bit of experience one simply knows them. Someone with a good intuitive feel for compactness would probably guess that they’re true, but I’d guess that almost everyone learns the standard results before developing that good a feel for the property.

The example that I suggested for (1) came from a basic strategy for approaching any result when you don’t know whether it’s true or not: look at some simple examples.

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  1. is not true. Consider $\mathbb{X} = [0, 1] \cup [2, 3]$ with the Euclidean distance and $f$ that maps $[0, 1]$ to $[2, 3]$ and $[2, 3]$ to $[0, 1]$.

  2. is a standard result in Topology. Every continuous function on a compact set is uniformly continuous.

  3. Closed subsets of compact spaces are compact. Compact subsets of metric spaces are closed. Put these together.

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