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### 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 ...
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 ...
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
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. ...
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 ...
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 ...
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### 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 ...
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 ...
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 ...
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 ...