Linked Questions

1
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1answer
1k views

How do i show that if every continuous function on $X$ is bounded, then $X$ is compact? [duplicate]

Let $(X,d)$ be a metric space. Assume every continuous function on $X$ is bounded. Prove that $X$ is compact. Well, i don't know which continuous function should i fix to start an argument. I tried ...
3
votes
1answer
225 views

Existence of maximizer implies compact? [duplicate]

I know that compact sets imply the existence of a maximizer, but is the converse true: Let $(X,d)$ be a metric space. Suppose that whenever $f$ is a continuous (and real) function on $X$, there ...
2
votes
1answer
128 views

A strange criterion for compactness [duplicate]

Is it true that if every continuous real-valued function on a metric space is bounded, then that metric space is compact
5
votes
1answer
51 views

Proving $K$ is compact directly. [duplicate]

If $K$ is a subset of metric space $\mathbb{R}^n$ and if every real valued continuous function on $K$ is bounded, then $K$ is compact. I know a proof considering $K$ is unbounded and not closed. ...
2
votes
1answer
56 views

Pseudocompactness implies Compactness in metric spaces [duplicate]

Let $X$ be a metrizable space. I'd like to prove that if $X$ is pseudocompact, then $X$ is compact (the converse is true by the Heine-Borel theorem). Suppose $X$ was not compact. Since $X$ is ...
0
votes
1answer
41 views

True /False question about Compact sets and continuous functions [duplicate]

I am trying some assignment questions and I am unable to think on how can I solve this problem. Question: Let K be subset of $\mathbb{R^{n} }$ such that every real valued continuous function on K is ...
1
vote
0answers
44 views

Proof that if $(X,d)$ is a metric space and $\forall f:X\to \Bbb R, (f\text{ continuous} \implies f \text{ is bounded})$ then $X$ is compact [duplicate]

Is my proof of this correct? We will prove the contrapositive, that if $X$ is not compact there exists a continuous real valued function on $X$ that is unbounded. Since $X$ is not compact, there ...
2
votes
3answers
160 views

Let $X \subseteq \Bbb Q^2$. Suppose each continuous function $f:X \to \Bbb R^2$ is bounded. Then $X$ is finite.

True or false: Let $X \subseteq \Bbb Q^2$. Suppose each continuous function $f:X \to \Bbb R^2$ is bounded. Then $X$ is finite. Now it will be compact for sure just by using distance function. Now ...
2
votes
2answers
435 views

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

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 ...
6
votes
2answers
243 views

Does “ All continuous functions are bounded ” or “ All continuous functions attain a maximum ” or together imply the domain is compact?

Let $(X,d)$ be an infinite metric space satisfying H1 or H2 or both. H1: (All continuous functions on X to $\mathbb{R}$ are bounded.) If $f: X\to\mathbb{R}$ is continuous on $X$, then $f(x)$ is ...
3
votes
1answer
627 views

Which of the following condition implies that the set $A$ is compact

Question : Let $A$ be a subset of $\mathbb R$. Which of the following properties implies that $A$ is compact $?$ Every continous function $f :A \rightarrow \mathbb R $ is bounded. Every sequence $ ...
1
vote
0answers
904 views

If every real-valued continuous bounded function on a metric space $M$ attains its maximum (or minimum), then $M$ is compact

Suppose that $(M,d)$ is a metric space. I want to show if every continuous bounded function $f:M \rightarrow \mathbb{R}$ achieves a maximum or minimum, them $M$ is compact. I found a similar ...
4
votes
0answers
566 views

Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
0
votes
1answer
226 views

Let $M$ be a metric space. If every continuous function $f : M \rightarrow \mathbb{R}$ has compact range then $M$ is compact. [duplicate]

Let $M$ be a metric space. If every continuous function $f : M \rightarrow \mathbb{R}$ has compact range then $M$ is compact. I got this question from "Real Mathematical Analysis" by Charles Chapman ...
0
votes
2answers
169 views

$X$ is metric space s.t. for every metric space $Y$ and any continuous function $f : X \to Y$ , $f(X)$ is closed in $Y$ ; is $X$ compact?

Let $X$ be a metric space such that for every metric space $Y$ and any continuous function $f : X \to Y$ , $f(X)$ is closed in $Y$ , then is $X$ compact ? Compare with this $A \subseteq \mathbb R^n $...

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