The compactness tag is for questions about compactness and its many variants (e.g. sequential compactness, countable compactness) as well locally compact spaces; compactifications (e.g. one-point, Stone-Čech) and other topics closely related to compactness.

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The set $\{\|f\|_\alpha \leq 1 \}$ has compact closure in $C([0,1])$

Recall the Holder norm $(0<\alpha\leq 1) $ $$\|f\|_\alpha = \max\bigg\{ |f(x)| + > \frac{|f(x) - f(y)|}{|x-y|^\alpha} : x,y \in [0,1], x\neq y\bigg\}$$ I want to show that the set ...
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1answer
29 views

Compactness of the Grassmannian $G(k,n)$

Related to this question, suppose we define $G(k,n)$ to be the set of $n\times k$ matricies with rank $k$, equipped with the quotient topology of $\mathbb{R}^{nk}$ by the equivalence relaiton $$A\sim ...
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1answer
140 views

Is there a structure theorem for nonempty, compact, nowhere dense subsets of the real line? [closed]

Let $X$ be the set of all nonempty compact nowhere dense subsets of the real line. Is there a theorem that describes the form of the elements of $X$? Context For open subsets of the line, such a ...
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1answer
27 views

Two definitions of compact set

I am reading parallely two books on analysis, and they have two different definitions of compact set: 1) Subset A of metric space X is called compact, if every open cover of A contains a finite ...
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3answers
37 views

Local compactness is preserved under continuous open onto mappings

If $f$ is a continuous open mapping of a locally compact space $(X,\tau)$ onto a topological space $(Y,\tau_1)$, show that $(Y,\tau_1)$ is locally compact. The definition of locally compact is ...
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3k views

A subset of a compact set is compact?

Claim:Let $S\subset T\subset X$ where $X$ is a metric space. If $T$ is compact in $X$ then $S$ is also compact in $X$. Proof:Given that $T$ is compact in $X$ then any open cover of T, there is a ...
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1answer
31 views

A countable, compact KC-subspace of a hereditarily Lindelöf minimal KC-space

A space in which all compact subsets are closed is called KC-space. A space in which every infinite set contains an infinite subset with only a finite number of accumulation points is said to have ...
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2answers
57 views

Showing that a space is normal and not locally compact

Let $E$ be the set of all ordered pairs $(m,n)$ of non-negative integers. Topologize $E$ as follows: For a point $(m,n)\neq (0,0)$, any set containing $(m,n)$ is a neighbourhood of $(m,n)$. A set ...
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2answers
361 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
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1answer
42 views

A continuous bijection from a Hausdorff space to a non-compact space which is not a homeomorphism

Recall the following theorem: Let $X$ be a compact space and $Y$ a Hausdorff space. Suppose that $f:X \rightarrow Y$ is a continuous bijection. Then f is homeomorphism. Prove that the ...
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50 views

Infinite spaces in which all subsets are compact are not Hausdorff

Let $(X,\tau)$ be an infinite topological space with the property that every subspace is compact. Prove that $(X,\tau)$ is not a Hausdorff space. I start by supposing $X$ is Hausdorff. Then I can ...
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2answers
61 views

Non-compactness of $\mathbb{R}$ with the cocountable topology

Is $(\mathbb{R},\tau_{co})$ compact where $\tau_{co}$ is the cocountable topology on $\mathbb{R}$? I have the answer of my teacher but I'd like to see another one so I can understand better how ...
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0answers
39 views

*Continuous real function on a compact set is bounded and achieves greater value and less in that set.* [closed]

Continuous real function on a compact set is bounded and achieves greater value and less in that set. I know how prove is done and how boundeness achieve greater value, please help prove someone as ...
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1answer
275 views

Characterization of compactness in weak* topology

Let $ X $ be Banach space, and $X^*$ its dual. A set $ F \subset X ^ * $ is weakly-* compact if and only if $ F $ is closed in the weak* topology and is bounded in norm. How does one prove this ...
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1answer
45 views

Boundedness of continuous functions on compact sets

Let $E$ and $F$ be two metric spaces. If $K$ is a compact subset of $E$ then a continuous function $f:K\to F$ is always bounded and reachs its maximum. What happens if we replace $K$ by a closed ...
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1answer
30 views

Show that a finite union of compact subspaces of a topological space $X$ is compact.

I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Can someone please verify my proof or offer suggestions for improvement? Show that a finite ...
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1answer
101 views

Can a function have a strict local extremum at each point?

A problem given in Spivak's Calculus text is to show that a function $f:[a,b]\to \mathbb{R}$ cannot have a strict local maximum at each point. I will sketch the proof below the fold. My question is: ...
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2answers
67 views

If a property holds for arbitrary compact set in a metric space, does it also holds for the metric space?

Suppose a metric space $(X, d).$ Further suppose that a property $A$ holds for arbitrary compact subset of $X.$ Does the property $A$ also hold for $X$? Context I hoped for some general theorems of ...
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3answers
170 views

In a non-Hausdorff space, can a compact subset fail to be closed?

In a Hausdorff space $X$, every compact subset $Y$ is closed. So if I relax the condition on $X$ being Hausdorff, is it possible compact subset $Y$ of $X$ not being closed?
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1answer
34 views

Proof of compactness for sets of norm equal to one in finite-dimensional normed vector spaces

The proposition I have been trying to prove is that the set $A=\{x\in E:N(x)=1\}$ is a compact subset of the (real) finite-dimensional vector space $E$ for any norm $N:E\to \mathbb{R}$. I am reading ...
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1answer
51 views

Does the converse of Tychonoff's theorem hinge on the axiom of choice?

Tychonoff's theorem:$\phantom{---}$ If $A$ is a non-empty index set and $X_{\alpha}$ is a non-empty compact topological space for every $\alpha\in A$, then $X\equiv\times_{\alpha\in A} X_{\alpha}$ is ...
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In a normal family, for every $\epsilon>0$ there are finitely many functions $f_{j}$ such that $\min_j\sup|f-f_j|<\epsilon$ for every $f$

Let $f_{n}$ be a normal family. Why does there exist finite many indices $f_{n_{1}}, \ldots, f_{n_{k}}$ such that $\{f_{n}: n = 1, 2, \ldots\} \subset \bigcup_{j = 1}^{k}\{f: |f(z) - f_{n_{j}}(z)| ...
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2answers
64 views

Clarifications on proof of compactness theorem

I've been reading through the following proof of compactness theorem: http://www.princeton.edu/~hhalvors/teaching/phi312_s2013/compactness.pdf One thing that struck me is that this proof seems to ...
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2answers
53 views

A question on the purpose of the condition on hausdorff to prove homeomorphism

This is a theorem proved in Munkres. Let $f:X\to Y$ be a bijective continous function. If X is compact and Y is hausdorff, then f is a homeomorphism. I knew Y being hausdorff which will be good to ...
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1answer
49 views

Compact closure in $C([0,2])$

a) Does the closure of $\left\{f_n(x)=\sin(x^n):n=1,2,3\dots\right\}$ form the a compact subset of $C([0,2])?$ b) Does the closure of $\left\{f_n(x)=\sin(x^\frac1n):n=1,2,3\dots\right\}$ form the a ...
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1answer
42 views

Proving that a quotient space is compact but not Hausdorff

Let ∼ be the equivalence relation on $\mathbb{R^2}$ defined by $(x, y) ∼ (x_0 , y_0 )$ if and only if there is a nonzero $t$ with $(x, y) = (tx_0 , ty_0$ ). Prove that the quotient space ...
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3answers
86 views

Confusion in proof of theorem ($2.7$) in Rudin's Real and complex analysis

I am not able to fill the gap in proof of following theorem which is stated as... Let $U$ be an open set in a locally compact hausdorff space $X$, $K\subset U$ and K is compact. Then there exists an ...
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4answers
138 views

Fixed point for a continuous function on a compact set?

If $f:X \rightarrow X$ is continuous and X is compact, will $f$ have a fixed point? We know that a contraction will have a fixed point but I have not come across an example of a continuous function ...
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5answers
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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 = \left\{0, 1, \frac{1}{2}, \frac{1}{3}, \dots , \frac{1}{n}, \dots\right\}\text{ and } B = ...
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What are some compact (Hausdorff) groups?

I just realized today that I don't know any compact groups that aren't profinite groups or Lie groups. Generalizing from these, a product of compact groups is again a compact group, a closed ...
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2answers
256 views

Prove or give a counterexample to the following converse of theorem: A continuous function on a compact set K(subset R) is uniformly continuous.

I think the converse of this theorem is: if every continuous function over $K$ is uniformaly continuous, then $K$ is compact. To find a counterexample of it, I want to show there exist a continuous ...
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58 views

Sequential compactness in $\mathbb{R}$

Well known result: Suppose $f:\mathbb{R}\to \mathbb{R}$ is continuous and let $K$ be a compact set. Then, $f(K)$ is compact. I can prove this using the definition of compactness (finding a ...
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alternative Compactness theorem proof

I'm attempting a problem which requires me to prove the compactness theorem for propositional logic ![enter image description here][1]in a slightly different way to normal. I'm struggling to ...
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17 views

Prove that a continuous one-to-one function from a compact space onto a hausdorff space is a homeomorphism [duplicate]

Some lecture notes I'm reading use the following lemme: let $ f : X \to Y$ be a continuous one-to-one function from a compact topological space $X$ onto a hausdorff space $Y$. Then $f$ is a ...
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2answers
34 views

Compact sets and Kuratowski limit

I have been struggling with the following claim: Let $A_n$ be a sequence of compact sets and $A$ a compact set. $A=\lim\sup_n A_n=\lim\inf_n A_n$ iff $d_H(A_n,A)\to 0$ where $d_H(.,.)$ is the ...
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1answer
36 views

Alexandroff one-point compactification = Freudenthal end point compactification / set of ends

By spaces I mean locally compact, $\sigma$-compact, connected, locally connected, Hausdorff topological spaces. I need a reference (not a proof, I already have one -- or at least I think so; ...
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1answer
108 views

The Arzelà–Ascoli theorem fails on a half-open interval

Can we find an example: (1) $\lbrace f_n \rbrace_n$ is a family of real-valued functions defined on $[0,1)$ such that this family is uniformly bounded and equicontinuous, $f_n(0)=0$; ~~~ Uniformly ...
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3answers
102 views

Compactness under different metric?

Consider the metric $\rho(x,y)=\frac{|x-y|}{1+|x-y|}$ on $\mathbb{R}$. Is $(\mathbb{R},\rho)$ compact? In order to show that is not, I wanted to find a sequence such that any subsequence is ...
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0answers
31 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 $X_2$ ...
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1answer
83 views

Suppose $K$ is compact. What other types of coverings must have finite subcovers?

Let $X$ be a topological space. Call $S\subseteq X$ an $\mathcal{O}$-set if there exists an open set $O$ such that $O\subseteq S \subseteq \overline{O}$. Suppose $X$ is compact. Is it true that any ...
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1answer
35 views

Compactness and Arithmetic Confusion

Let $T$ be some theory capable of arithmetic and construct a provability predicate (which we will call $Prb_T$). Let $\mathbb{N} \models T$. Expand our language to include a new constant symbol $c$. ...
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1answer
50 views

Choice of number in the proof the 5-r covering theorem

Why has the number 3 been chosen? I have tried drawing this and it seems wrong (its not). The balls definitely dont seem to be disjoint either. It would seem that if a particular $x$ has $r(x)$ ...
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1answer
195 views

Stone Cech compactification homeomorphism implies realcompactification homeomorphism

I was wondering: If $\beta X$ is homeomorphic to $\beta Y$, is it true that $\nu X$ is homeomorphic to $\nu Y$? Notation: If $f: X\rightarrow \mathbb R$, we denote it's extension by $f^\alpha: \beta ...
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3answers
225 views

For a compact covering space, the fibres of the covering map are finite.

I am stuck on the following exercise: Let $Y$ be a compact topological space, and $p:\ Y\ \longrightarrow\ X$ a covering map. Show that for every $x\in X$ the fibre $p^{-1}(x)$ is finite. Any ...
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2answers
87 views

Continuous function from R to a compact set

I know that a continuous function maps compact sets into compact sets. My question now is, are there continuous functions $f:{\mathbb R}\rightarrow I$, with $I=[a,b]$ ($a\neq b$)?
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1answer
34 views

zero-sets of $\beta X$

I'm trying to understand the following proof from Walker: Proposition. The zero-sets of $\beta X$ are countable intersections of closures in $\beta X$ of zero-sets of $X$. Proof. If $Z$ is a zero ...
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16 views

Converse to sequential Banach--Alaoglu [duplicate]

Let $B$ be the closed unit ball of the dual space of a real normed vector space $V$. If $V$ separable then $B$ is sequentially compact in the weak-* topology. What about the converse?
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3answers
165 views

The metrizable space may be not locally compact

My text book said: Not every metrizable space is locally compact. And it lists a counterexample as following: The subspace $Q=\{r: r=\frac pq; p,q \in Z\}$ of $R$ with usual topology, i.e., ...
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1answer
43 views

Prove that a set is compact

Let $X$ be a compact space, let $U$ be an open set in $X$, Let $f:U\to [0,1]$ be a continuous map. Prove that the set $$K=\{(x,t): x \in U , 0 \leq t \leq f(x) \} \subset X \times [0,1]$$ is compact. ...
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1answer
59 views

A closed bounded set having exactly one accumulation point has the covering property

I need to show that a closed, bounded set having exactly one accumulation point has the covering property. A set has the covering property if any open cover of it has a finite subcover. Since the ...