3
votes
2answers
90 views

Proving a set is compact - Homework

Let $(X,d)$ be a metric space and let {$p_n$} be a sequence of points in $X$ with $\lim_{n\to ∞}p_n = p_0$. Prove that the set $K =$ {$p_0, p_1, p_2,...$} is a compact subset of $X$. I have ...
2
votes
1answer
25 views

Continuous function on $\mathbb{R}^{n}$ preserving compactness - some clarification

My professor went over a proof of the following in class: Suppose $A \in \mathbb{R}^{n}$ is compact and $f:A \rightarrow \mathbb{R}^{n}$ is continuous. Then $f(A)$ is compact. The proof ...
1
vote
1answer
28 views

Compactness of the convergent to zero sequences

I've gotta prove that $$T = \left\{ \left\{ x_i \right\} \in {\ell ^\infty }:\left| x_i \right| < \mu_i,\mathop \lim\limits_{i \to \infty } \mu _i = 0 \right\} \subseteq \ell ^\infty $$ is ...
0
votes
0answers
35 views

Compactness criterion

I have this compactness criterion and I want to apply it, but I don't know what I must write to see if (a) is satisfied and also for (c)? For a subset $H\subset\mathcal{BC}(\mathbb{R},Y)$ to be ...
2
votes
1answer
23 views

Counterexample on weaker version of result about compact sets

The following is a very well known theorem: Let X be a metric space. $K \subset X$ is compact iff every collection $ \{ F_j \}_{j\in A}$ of closed sets with the finite intersection property in K ...
1
vote
1answer
54 views

Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
0
votes
0answers
28 views

Definition of a Paracompact space

I have a question about the definition of a paracompact space. We said that a space $X$ is paracompact iff $X$ is $T_2$ and if any open covering of $X$ has a finer locally-finite covering. I don't get ...
0
votes
3answers
69 views

Proof that the continuous image of a compact set is compact [duplicate]

Let $X\subset \mathbb R^{n}$ be a compact set, and $f :\mathbb R^{n}\to \mathbb R $ a continuous function. Then, $F(K)$ is a compact set. See, I know that this question may be a duplicate, but the ...
0
votes
0answers
21 views

Jordan content under continuous differentiable map

I have the following problem which seems simple but in fact I find no proof for it so I am wondering if I could get some help. Let $A$ be a compact set subset of an open set $U$ in $\mathbb{R^n}$, ...
3
votes
2answers
87 views

Direct proof of compactness of $\mathbb{Z}_p$

Let $\mathbb{Z}_{p}$ be completion of $\mathbb{Z}$ with respect to $p-$norms. Actually I know that $\mathbb{Z}_{p}$ is bijective to Cantor set, which is compact, therefore by homeomorphism, it is also ...
2
votes
0answers
88 views

Proof of uniform continuity on compact sets

Show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous on a compact set $K$ is uniformly continuous on $K$. Is the proof below correct? Proof: Let $\epsilon > 0$ and let ...
4
votes
2answers
52 views

$\left\{x\in H: 2\leq \|x\|\leq 5\right\}$ is compact?

In a Hilbert space $H$ of dimention infinite, $A=\left\{x\in H:2\leq \|x\|\leq 5\right\}$ is compact? (totally bounded and complete) Thanks in advance.
1
vote
1answer
18 views

Continous function on compact interval - bounded

Let $K$ be a compact interval in $\mathbb{R}$. Then every continous function $\phi :K\rightarrow \mathbb{R}^d$ is automatically bounded. Is this a consequence of; the image of a compact is compact ? ...
0
votes
0answers
99 views

understand proof of compactness in product topology

I am trying to understand the following reasoning. Call $\mathcal{F_\lambda}$ the set of functions $a:\mathbb{N} \to \mathbb{R}$ for which $Na(i) := \sum_{j \in \mathbb{N}} n_{ij} a(j)\leq \lambda ...
2
votes
1answer
79 views

compactness in topology of pointwise convergence

I started reading about the topology of pointwise convergence. So far I do not feel quite comfortable with this theory. Maybe one can help me out in a more concrete example case. Let's consider ...
1
vote
3answers
136 views

Closed subset of compact set is compact

If S is a compact subset of R and T is a closed subset of S,then T is compact. (a) Prove this using definition of compactness. (b) Prove this using the Heine-Borel theorem. My solution: ...
5
votes
1answer
80 views

Closure of compact sets in Banach space

Let $(X,\vert\vert\cdot\vert\vert)$ be a Banach space. For each $k\in\mathbb{N}$ let $A_k\subseteq X$ be compact and $r_k\in\mathbb{R},r_k>0$, such that $$A_{k+1}\subseteq \{x+u\vert x\in A_k ...
6
votes
2answers
120 views

Compactness and closedness

If every closed and proper subset of a topological space $X$ is compact, then is the whole space necessarily compact? The "converse" of this question is well-known, of course, but I'm having ...
2
votes
2answers
91 views

Compact Domain and Inverse Image

I am trying to show that given $f:M \rightarrow N$, where $M$ is compact, $f$ is continuous and onto, then given $A \subset N$: $$ f^{-1}(A) \text{ closed} \implies A\text{ closed} $$ I am dealing ...
1
vote
2answers
262 views

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 ...
3
votes
1answer
47 views

Analysis question.

Is this set compact? $\{(x,y) \in \mathbb R^2 : |x|+|y|\leq 1\}$. I know that is closed and bounded so compact but I don't know how to show it is closed and bounded mathematically. This is the graph ...
0
votes
0answers
128 views

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 ...
2
votes
1answer
364 views

In a Hilbert space, every bounded and closed set is weakly relatively compact.

My aim is to prove that in a Hilbert space, any sequence has a weakly convergent subsequence. To prove this, I'm trying to prove that: ...
0
votes
3answers
61 views

Is the $\omega$-product of the set of irrationals compact?

We know that any product of compact spaces is compact. But, I wonder that the countable product of $\mathbb{P}$ can be compact since $\mathbb{P}$ is not compact?
0
votes
0answers
52 views

Is an epsilon-net dense in its totally bounded set?

By definition, a totally bounded set A possesses an epsilon-net for every epsilon greater than 0. Does this mean that every point of A is either a limit point of the epsilon-net or a point in the net? ...
2
votes
1answer
45 views

Does the union of all these neighborhood cover $[0,1]$

Consider the rationals in $[0,1]$. Around each I take a neighborhood (possibly of different radii). Is the union of all these neighborhood sure to cover $[0,1]$? What if I had used irrationals instead ...
0
votes
1answer
44 views

Uniformly continuous on a compact set, still uniform on a subset?

So if I have a function that is uniformly continuous on a compact set K, do all subsets of K inherit the uniform continuity? If I restrict myself to the reals, this seems to be true. But what happens ...
2
votes
1answer
76 views

Proving completeness and compactness of a sequence of metric spaces.

The problem statement Let $(X_n,d_n)_{n \in \mathbb N}$ be a sequence of metric spaces. Consider the product space $X=\prod_{n \in \mathbb N} X_n$ with the distance $d((x_n),(y_n))=\sum_{n \in ...
2
votes
1answer
43 views

Proving two statements about locally compact spaces

The problem statement: Let $(X,d)$ be a locally compact metric space (for every $x \in X$, there exists a compact neighbourhood of $x$) $a)$ Prove that if $K_1 \subset X$ is compact, then, there are ...
0
votes
1answer
35 views

Proving the set of “distance functions” on a compact set is a compact set itself

The problem statement. Let $(X,d)$ be a compact metric space and $C(X)=\{\phi: X \to \mathbb R : \phi \text{ is continuous}\}$. For each $x \in X$ we define the function $f_x: X \to \mathbb R$ ...
1
vote
1answer
38 views

Demonstrate that the following metric space is not compact

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 $X$ is not compact. I know that sequentially compact and ...
0
votes
1answer
24 views

Help Understanding: Closed Subspace of Compact Space is Compact on ProofWiki

http://www.proofwiki.org/wiki/Closed_Subspace_of_Compact_Space_is_Compact ProofWiki provides the following proof that a closed subspace of a compact space is compact: Let $T$ be a compact space. Let ...
2
votes
1answer
60 views

Prove or disprove a set $F$ is closed.

This is an example in my book that talks about $F$ being precompact; Let $F$ be the subset of $C([0,1])$ that consists of functions $f$ of the form $$f(x) = \sum_{n=1}^{\infty}a_n\sin(n\pi x) ...
1
vote
1answer
77 views

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 ...
1
vote
1answer
124 views

Compact Subsets of $C[a,b]$

Consider the set $G = \lbrace f \in C\left[a,b\right] : |f(x)| \le |g(x)|,\ \forall x \in [a,b] \rbrace$ Find all values of $g$'s for which $G$ is a compact subset of $C[a,b]$ with the max norm. ...
2
votes
1answer
233 views

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 ...
2
votes
2answers
91 views

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 ...
1
vote
1answer
297 views

Questions about coercive functions and its implications

Given this definition: A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is $coercive$ if $$\lim_{||x||\rightarrow\infty}f(x) = \infty.$$ Explicitly, this means that for any $M>0$ there is an ...
3
votes
1answer
225 views

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum.

Let $X$ be a compact metric space. If $f:X\rightarrow \mathbb{R}$ is lower semi-continuous, then $f$ is bounded from below and attains its infimum. I want to prove this. This is my proof: Since $X$ ...
1
vote
1answer
267 views

A question about Hausdorff locally compact spaces

In one of the proofs of Rudin's real complex analysis, the following implication seems to be assumed: Let $X$ be a locally compact Hausdorff space. Let $V$ be an open subset of $X$. Let $x\in V$. ...
0
votes
1answer
57 views

Coercivity and Compactness

My question is: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be continuous on all of $\mathbb{R}^n$. Prove $f$ is coercive and and only if for every $\alpha\in\mathbb{R}$ the set $\{x|f(x)\leq\alpha\}$ ...
67
votes
12answers
5k views

Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel ...
0
votes
2answers
84 views

Compactness of the set $(0, 1)$ as a subset of $\mathbb{R}^2$?

An open ball in $\mathbb{R}^2$, centered at the point $(1/2, 0)$ and of radius $1/2$ covers the segment $(0,1)$. The open ball thus forms a finite cover of $(0,1)$, implying that $(0,1)$ is a compact ...
3
votes
1answer
456 views

Compactly supported continuous function is uniformly continuous

Let $f:\mathbb R \rightarrow \mathbb R$ be continuous and compactly supported. How can I prove that $f$ is uniformly continuous ? I was trying to prove it by contradiction but get stuck. My attempt ...
1
vote
1answer
192 views

Lebesgue's criterion for Riemann integrability of bounded real valued functions defined on compact metric spaces

Let $(X, d)$ be a compact metric space, and let $S$ be the algebra of sets generated by the open and closed balls of $X$. Suppose we have a pre-measure defined on $S$ such that the measure of each ...
0
votes
1answer
132 views

Compactness and compact-finite measure in Lusin theorem (Rudin)

I have two questions about some hypotheses in Lusin's theorem as stated in Rudin's "Real and Complex Analysis". The proof initially deals with a subcase, that is the function $f$ is supposed to be ...
1
vote
1answer
93 views

Closed set in $l^1$ space

Let $$ X := \left \{ (a_n) : \sum_{n=0}^\infty |a_n| < \infty \right\}$$ with the metric $d(a_n,b_n) := \sum_n |a_n-b_n|$. Let $\delta_j^{(n)} := 1$ if $n = j$ and $0$ otherwise. Denote ...
2
votes
1answer
243 views

The set of all subsequential limits of a bounded sequence is a non-empty compact set

Let $(x_n)$ be a bounded sequence and let $Y$ be the set of all subsequential limits of $(x_n)$. Prove that $Y$ is a non-empty compact set. I think it's possible to solve this problem by proving that ...
3
votes
1answer
302 views

Product of two compact topological spaces is compact.

Proof: I am following this. But, I feel, I am missing something. Consider two compact spaces $X_1$ and $X_2$ and some cover $U$ of their product space. Consider an element $x\in X_1$. The sets ...
1
vote
2answers
137 views

Hausdorff space and Cantor's intersection theorem

$X$ is a Hausdorff space, $C_i$ is a non-empty closed subset of $X$ and $C_{k+1}\subseteq C_k$ , show that $\displaystyle \bigcap_{i\in \mathbb{N}} C_i$ is compact. I tried to prove by ...