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What is the relation between compactness, connectedness and continuous real-valued functions on $\mathbb{R}^n$, n > 1?

For example, 1- what is the relation between a compact set and boundedness of every continuous real valued function on it?

2- If a set on $\mathbb{R}^n$ is compact, is it bounded? and if it is bounded is it compact?

Emptymind
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2 Answers2

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Suppose $X$ is compact and $f:X\to\mathbb{R}^n$ is continuous. Then $f$ is bounded.

Proof: Since the continuous image of compact spaces are compact, we know that $f(X)$ is a compact subset of $\mathbb{R}^n$. Thus it is a bounded subset of $\mathbb{R}^n$ by the Heine-Borel theorem.

The Heine-Borel theorem mentioned above says:

A subset $E$ of $\mathbb{R}^n$ is compact iff $E$ is closed and bounded.

Thus compact subsets of $\mathbb{R}^n$ are bounded. However, the converse is not true. For instance, $(0,1)$ in $\mathbb{R}$ is bounded but not compact.

John Griffin
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  • Is it true that "If every continuous real valued function defined on a nonempty subset of $\mathbb{R^n}$", n>1, is bounded, then this non empty subset is compact? – Emptymind Aug 20 '17 at 23:53
  • @Intuition Yes. – John Griffin Aug 20 '17 at 23:58
  • why ? ..... is there a theorem that said this? – Emptymind Aug 22 '17 at 07:29
  • I have a problem ..... your answer contains half of the answer of the question and @Marios Gretsas contains the other half ..... so I do not know who to accept? – Emptymind Aug 22 '17 at 07:32
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    @Intuition accept whatever you want..accept the answer that you believe it helps you the most..Don't worry.. If you accept John's answer,which of course is a nice answer,there would be no problem by me. – Marios Gretsas Aug 22 '17 at 12:18
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    @Intuition See here: https://math.stackexchange.com/questions/668905/if-every-real-valued-continuous-function-is-bounded-on-x-metric-space-then – John Griffin Aug 22 '17 at 14:19
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A continuous image of a connected set is connected.

Also a topological space $X$ is connected iff there does not exist a surjective continuous function from $X$ to $\{0,1\}$(or any set with two elements in the reals for instance).

Now for compactness-continuity we have also these results:

If $f:X \rightarrow Y$ is a continuous bijection and $X$ ia compact topological space and $Y$ is a Hausdorf topological space then $f$ is a homeomorphism.

.

Every continuous function on a compact metric space $X$ is uniformly continuous on $X$

Marios Gretsas
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  • can I say that if a subset of $\mathbb{R^n}$, where n > 1, is compact then its connected? – Emptymind Aug 20 '17 at 22:49
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    No because the set $([0,1] \times [0,1]) \cup( [5,6] \times [5,6])$ is compact but not connected..Note that a finite union of compact sets is compact. – Marios Gretsas Aug 20 '17 at 22:54
  • I have a problem ..... your answer contains half of the answer of the question and @John Griffin contains the other half ..... so I do not know who to accept? – Emptymind Aug 22 '17 at 07:31