# Tagged Questions

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### Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed

Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed I know that $A$ and $B$ are closed and bounded, then they are sequentially compact, so $A+B$ also sequentially compact, ...
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### The continuous image of a sequentially compact set is also sequentially compact.

Let $S$ be a sequentially compact set and let $f : S\to R$ be continuous. Then the image $f(S)$ is sequentially compact.
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### a compact set $X$ has a countable set $S$ such that $\overline{S} = X$

Suppose $X \subseteq \mathbb{R}^d$. Suppose $X$ is compact. Then there exists a countable subset of $X$, $S \subseteq X$ such that $\overline{S} = X$. How can I show this? I have no idea how to ...
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### If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.

Let $X$ be a metric space. Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact. This has been asked before, but all the answers I have seen prove the ...
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### Equivalence conditions in the Heine-Borel theorem for the real line

The Heine Borel theorem (book, pg 335) shows that the following conditions are equivalent- A set $K$ is closed and bounded. $K$ is compact. My question is that in the proof of 1 $\implies$ 2 where ...
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### Compactness implies Continuity?

I am stuck on this question (probably there are many counterexamples, but I can't find any). "Suppose $f:\mathbb{R}\mapsto\mathbb{R}$ that preserves compactness (i.e, for every $K \subseteq R$, then ...
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### Metric-space, counterexample in Arzela-Ascoli Theorem

My book has very few examples, so I would like an example covering this. The theorem is stated as follows. "Let $(X,d_{X})$ be a compact metric space. A subset K of $C(X,\Re^{m})$ is compact if and ...
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### Sequence of elements having a convergent subsequence -NBHM $2014$

Question is to find which of the following are true? Let $V$ be the space of continuous functions on $\mathbb{R}$ with compact support endowed with metric ...
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### Is the following set is compact

Consider the set of all $n \times n$ matrices with determinant equal to one in the space of $\mathbb R^{n\times n}$. My idea is compact because determinant function is continous ant it is bijective ...
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### Suppose that for all $t <1$ there are points $x_t$ and $y_t$ such that $d(x_t,y_t) = t$.

Let $(X,d)$ be a compact metric space. Suppose that for all $t <1$ there are points $x_t$ and $y_t$ such that $d(x_t,y_t) = t$. Prove that there exists points $x$ and $y$ such that $d(x,y) = 1$. I ...
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### Prove that :- If K is a compact subset of R with non empty interior then it is of the form [a,b] or [a,b] - U{In}

The question is :- Let $K$ be a compact subset of $\mathbb R$ with non empty interior. Show that K is of the form $[a,b]$ or $[a,b] \setminus \bigcup I_n$ , where { $I_n$} is a countale disjoint ...
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### Show that the set $K=[0,1]$ is compact in $\Bbb{R}$

Please help me for the follow example. Please. The example is: Show that the set $K=[0,1]$ is compact in $\Bbb{R}$. Thanks for your solve. Thanky very much once again for your answers and your ...
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### closed and bounded but not compact set of real-valued bounded functions

I'm trying out a problem I was given and this is the statement: Prove, or disprove, that every bounded and closed subset of the set of real-valued and bounded functions on [0,1] equipped with the sup ...
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### Compact subsets of a metric space

I am trying to to prove that f: X --> Y is continuous on X if and only if f is continuous on every compact subset of X. X and Y are metric spaces. How do I show that every point of X belongs to some ...
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### Proving that a Closed Interval Is Compact

My text (Stoll, Introduction to Real Analysis, 2nd Ed) defined that $K$, a subset of $\mathbb R$, is compact if every open cover of $K$ has a finite subcover of $K$. Then, it proceeded to prove that ...
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### Show that $A= ([0,\sqrt2] \cap \Bbb Q ) \subset \Bbb Q$ is not compact.

We have $\Bbb Q$ equipped with the Euclidean Metric. Show that $A=([0,\sqrt2] \cap \Bbb Q ) \subset \Bbb Q$ is not compact. How would you go about showing this? You can make on open cover ...
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### Analytic function on a compact set

I am trying to prove the following but I am not sure how to use the compactness: Let $f$ be a real analytic function on a compact set $K$. Let $(x_n)_n$ be a sequence in $K$ such that $f(x_n) = 0$. ...
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### Arzela-Ascoli net question

Let $X$ be a compact metric space. Let $C(X)$ denote the space of real-valued continuous functions on $X$. A commonly given corollary to the Arzela-Ascoli theorem is: Proposition: If $f_n$ is an ...
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### Real Analysis: Compact Sets

I'm working on a general real analysis problem involving compact sets. I was given these two sets: A={0,1,1/2,1/3, ... ,1/n, ...} and B={1,1/2,1/3, ... ,1/n, ...} I'm supposed to figure out which ...
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### Compactness of $x^4+y^4=1$

Show $A=${$(x,y):x^4+y^4=1$} is compact. So far, I'm thinking I should mention that $0 \le x \le y-x \le 1$ and $0 \le y \le x-y \le 1$ determines the values of $(x,y)$, and since [0,1] is compact, ...
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### Compactness of a set in a Locally compact Hausdorff space

My question is about the following lemma: Let $X$ be a locally compact Hausdorff space, and $V$ be a nbhd. of a point $x \in X$. Then there is a nbhd. $U_{x}$ of $x$ such that $\overline{U_{x}}$ is a ...
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### $X$ normed linear space separable $\Longleftrightarrow$ $\exists K \subset X$ compact s.t. $\overline{ \text{span}\{K\}}= X$

Let $X$ be a normed linear space. Show that $X$ is separable if and only if there is a compact subset $K$ of $X$ for which $\overline{ \text{span}\{K\}}= X$ I can't figure out how to solve this ...
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### About the properties of Measure of noncompactness

Let $\alpha$ denote the Kuratowski measure of noncompactness defined on the Banach space $(E,\|.\|)$ and $A, B\subset E$ be nonempty, bounded subsets. Then, how to prove that if $A\subset B$ then ...
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### Prove Heine-Borel Thm

Prove Heine-Borel Theorem: "A subset $S$ of $\mathbb{R}$ is compact if and only if every open cover for $S$ has a finite subcover." Suggestions: Let $S \subset \mathbb{R}$. If every open cover for ...
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### Show that the metric space is not compact

I have a proof that I would like some hints in solving: Let $X$ be a metric space. Show, if there is an $r > 0$ and a sequence $(x_n)$ from $X$ such that $d(x_n,x_m) \geqslant r$ for $n≠m$, then ...
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### Compactness of $[0,+\infty)$

Let's say we have $$F = [0,∞).$$ How can we determine whether this is a compact set or not? And let's say we have $U = {(-1,n)}$ ($n∈N$), book said that this $U$ is the open cover of $F$, but I ...
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### Proving that the Union of Two Compact Sets is Compact

Prove if $S_1,S_2$ are compact, then their union $S_1\cup S_2$ is compact as well. The attempt at a proof: Since $S_1$ and $S_2$ are compact, every open cover contains a finite subcover. Let the ...
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### Showing a set is not compact by describing an open cover that doesn't have a finite subcover

I would like to prove that the following set is not compact by stating an open cover for it that has no finite subcover. $E=\{x\in\mathbb{Q}:0\leq x\leq2\}$ I'm having trouble thinking of one. A ...
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### Show that the set $[0,2] \setminus \{1\}$ is not compact by exhibiting a cover of open intervals which has no finite subcover.

I think $\bigcup_{k=1}^\infty \left(\frac{-1}{k},2-\frac{1}{k}\right)$ will work, but I'm unsure if the interval includes $2$ as $k \rightarrow \infty$.
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### Show that $f$ is onto [duplicate]

Let $(X,d)$ be a compact metric space. Let $f:X\rightarrow X$ be such that $d(f(x),f(y))=d(x,y)$ for all $x,y\in X$. Show that f is onto. Hint: Fix $y\in X$and $x_1\in X$, define $x_n=f(x_{n-1})$, ...
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### Working in $\mathbb{R}^n$. Prove that there exists $x$ in $A$ and $y$ in $B$ such that $\mathrm{dist}(A,B)=\|x-y\|$.

Let $A$ be a compact subset of $\mathbb{R}^n$ and let $B$ be a closed subset of $\mathbb{R}^n$. Also suppose $A$ and $B$ are disjoint. Prove that there exists $x$ in $A$ and $y$ in $B$ such that ...
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### Real analysis, showing that a set is not compact.

I am limited to theorems from Rudin, so think basic real analysis. Given: $l^1$ - set of sequences such that the infinite series consisting of terms $|a_n|$ converges. i.e absolute convergence. ...
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### Uncountable open cover of $\mathbb{R}$

The question posed to me is to find an uncountable open subcover of $\mathbb{R}$ such that it has no finite subcover, but I can't even think of a way to define an uncountable open cover.
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### Closed and bounded does not mean compact in general

Suppose in metric space $\Bbb Q$ I consider the subset $\{x\in\Bbb Q | 2<x^2<3\}$. I want to find a open cover for this set which does not have any finite subcover. How to do it ? Hint is ...
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### Clarification on this corollary of the Arzela-Ascoli Theorem

I am given the following corollary without proof: A family of continuous functions on a compact metric space into $\mathbb R^m$ is compact iff it is closed, equicontinuous and bounded. Does ...
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### closed,bounded not compact

Hi I was asked to prove that: if $S =\{ x \in \Bbb R : d(x,0) = 1 \}$ then $S$ is a closed and bounded set. The set $S$ contains only two points: $-1,1$,(it should not be a problem to prove that is ...
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### What does it mean for a set to be compact in another set?

I am given the following definition: Let $B$ be a set of continuous maps with domain a metric space $A$ and codomain a metric space $N$, and $B_x=\{f(x):f\in B\}$. $B$ is pointwise compact ...
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### Does extreme value theorem hold when continuous is replaced with bounded?

The extreme value theorem says that if the domain of a 'continuous' function is compact then both the max and min of the function lies in the domain set. My question is: can the 'continuity' be ...
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### let $K \subset U \subset X$, $(X,d)$ metric space $U$ open and $K$ compact, prove the exist an $r>0$ such that $d(x,K)\leq r \rightarrow x \in U$

I'm stuck... I would appreciate some help let $K \subset U \subset X$, $(X,d)$ metric space $U$ open and $K$ compact, prove there exists an $r>0$ such that $d(x,K) \leq r \rightarrow x \in U$
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### Can this intuition give a proof that an isometry $f:X \to X$ is surjective for compact metric space $X$?

A prelim problem asked to prove that if $X$ is a compact metric space, and $f:X \to X$ is an isometry (distance-preserving map) then $f$ is surjective. The official proof given used ...
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### An Open Cover $\mathscr{F}$ of $2\mathbb{N}$ That Has No Finite Subcover

What is an open cover $\mathscr{F}$ for the set $2\mathbb{N}=\{2n:n\in\mathbb{N}\}$ that has no finite subcover? My initial answer is ...
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### $-\mathbb{N}$ is not bounded below, right?

Parable: I am asked to prove by contradiction that the set of negative integers $\mathbb{N}$ is not bounded below. My professor writes $$-\inf(-\mathbb{N})=\sup(\mathbb{N}),$$ and says that since ...
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### What is $\sup(\sin(n))$? [duplicate]

My classmate asked a question during lecture about our discussion of bounded sequences, particularly the sequence $\sin(n)$. His question was, What is $\sup(\sin(n))$?
From Rudin Theorem: Suppose $f$ is a continuous mapping of a compact metric space $X$ into a metric space $Y$. Then $f(X)$ is compact. The proof is outlined as follows Let $\{ V_\alpha\}$ ...
### $d(f,g)=\sup\{\lvert f(x)-g(x)\rvert : x\in[0,1]\}$ [closed]
Let $F$ be the non-empty set of functions that map from $[0,1]$ to $[0,1]$. Is $$d(f,g)=\sup\{\lvert f(x)-g(x)\rvert : x\in[0,1]\}$$ a metric on $F$?