Tagged Questions

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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Is $\{(x,y) \in \mathbb R^2 \mid x^2 + x^3y^2 = 0\}$ compact in $(\mathbb R^2, \mathcal E_2)$?

Determine if $X = \{(x,y) \in \mathbb R^2 \mid x^2 + x^3y^2 = 0\}$ is compact in $(\mathbb R^2, \mathcal E_2)$, where $\mathcal E_2$ denotes the standard Euclidean topology. I know that $X$ is ...
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The compactness of metric space $\mathcal{G}_n$

Here, metric space $\mathcal{G}_n$ is described below: It is said that it is well-known that $\mathcal{G}_n$ is compact for every $n$. But I can't find a proof, can you give me a proof or an ...
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Does being Nonempty Compact Set on $\mathbb{R^+_2}$ imply being Convex set?

Look at the domain of a function $y=x-2$ where $x\in\mathbb{R_+}$. Then, the triangle produced by x and y-intercepts is bounded and closed. So it is compact. Suppose it is also nonempty. Does this ...
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Generalization of Strict Local Maxima

I try to generalize a strict local maximum to a local roof which can possibly be a flat area instead of just a single point. Below is my attempt: Let $f$ be a continuous real-valued function on $R^D$ ...
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Question about the proof that the Hilbert Cube is compact.

Because of the fact that $(1)$ The topological space $[0,1]$ is a continuous image of the Cantor space $(G,T)$. There exists a mapping $\phi_n$ of $(G_n, T_n)$ onto $(I_n, T'_n)$ where, for each ...
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Uniform convergence of a equicontinuous sequence of functions [duplicate]

Let $X$ be a compact metric space, and $(C(X),d_{\infty})$ the space of continuous functions. Let $D\subset{X}$ be a dense subset, and $\{{f_n\}}_{n \in N}$ a equicontinuous sequence of functions from ...
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Checking the compactness of sets

I have to check to following sets for compactness in the given spaces with respect to the standard norm for them: \begin{align} M_1 &:= \{f_n:\left[-1, 1\right]\rightarrow \Bbb{R}| f_n(x) = n \cos(...
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Continuous map on $S^2$

Can you help me with this? Let $S^2 := \{x\in \mathbb R^3:||x||_2 = 1\} \subset (\mathbb R^3, ||\cdot||_2)$ and $T:S^2 \to (\mathbb R, |\cdot|)$ be a continuous map. a) Why does T assume its ...
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Show that a subset $E$ $\subset \Bbb Q$ is not compact in $(\Bbb Q, d)$ and decide whether it is open or not

Assume $(\Bbb Q, d),$ $d(p, q):= |p -q|$ is a metric space and $E := \{p \in \Bbb Q : 2 < p^2 < 3\} = \{p \in \Bbb Q : \sqrt2 < p < \sqrt3\} \subset \Bbb Q.$ I have to show ...
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On the matter ; If $f:X \to Y$ is a function with closed graph and compactness preserving then $f$ is continuous

Let $X,Y$ be metric spaces , $f:X \to Y$ be a function , with closed graph , carrying compact sets to compact sets ; then I claim that $f$ is continuous Proof: Let , if possible , $f$ be not ...
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Motivation and intuition behind concept of totally bounded-ness in metric spaces

The way I understand it is, a set in a m.s. is totally bounded means the set Admits a finite open cover of fixed size. Regardless of whether it admits an arbitrary open cover. Why do we need totally ...
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Is it true that the closed convex hull of a compact subset of the dual equipped with the w*-topology is compact?

Let $X$ be a Banach space. Consider the dual $X^*$ equipped with the weak*-topology. Is it true that the closed convex hull of a compact subset $K$ of the dual $X^*$ is compact? ps: I know that the ...
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Difficulty in choosing correct answer among the options.

1) The Cantor set, a subset of the real numbers: A. is not compact. B. is not contained in an interval. C. does not contain a non-trivial interval. D. does not have uncountably ...
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Locally compact Hausdorff and Second countable space and nested compact sequences

I have seen a theorem in my lecture notes without proof. It says: If $X$ is a locally compact Hausdorff and second countable space, then it can be written as a nested sequence of compact subsets ...
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Shift of first category set in a compact metric space.

Question. If $X$ is a homogeneous compact metric space, and $F=\bigcup _{n\in\omega}F_n$ is a countable union of closed nowhere dense subsets of $X$, then is there a homeomorphism $\varphi:X\to X$ ...
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Every sequentially compact space is countably compact

Every sequentially compact space is countably compact. The most that I can get out of this $(\Rightarrow)$ is that ${x_{n_j}} \subseteq \bigcup_{i=1}^{n_0}O_{x_i}$, where $O_{x_i}$ is open ...
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A topological space is countably compact iff every countably infinite subset has a limit point

A topological space is countably compact iff every countably infinite subset has a limit point. I'm completely stuck on this one. The book is recommending to use the fact that a space is countably ...
$(X,d)$ is a compact metric space, $\{U_i | i \in I\}$ is an open cover of $X$. I need to show that there is a number $\delta > 0$, such that if $A \subset X$ and diameter of $A \le \delta$, then ...
Let $X$ a Tychonoff space, $(e, \beta X)$ Stone-Cech Compactification and consider $(i,K)$ another compactification of $X$ that satisfies: $\forall f: X \to \mathbb R$ continuous and bounded, exists \$...