# Tagged Questions

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### non trivial convergent sequence

Let $\beta\omega$ be the Stone-Čech compactification of the natural numbers. We know that it is compact and Hausdorff, but it has no non-trivial convergent sequence. Is there an example else to be ...
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### How to prove that a first-countable topological space satisfies “closed-compact” if and only if it is compact Hausdorff?

A topological space is called $C$-$C$ iff the closed sets in $X$ coincide with the compact sets in $X$. Let $(X,\tau)$ be a topological space which satisfies the first axiom of countability. Then ...
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### Let $X$ be a $US$ space. Then $X^*$ is $US$ iff in $X$ , every convergent sequence has a relatively compact subsequence

$( X^*,\tau^*)$ is one - point compatification of topological space $(X, \tau)$. A topological space is called a $US$-space provided that each convergent sequence has a unique limit. The bellow ...
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### A sequential minimal KC-space is compact

A space $(X,\tau )$ is said to be minimal KC , if $(X,\tau )$ is KC but no topology on X which is strictly smaller than $\tau$ will be KC Theorem : A sequential minimal KC-space is compact. ...
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Accumulation point: Let $\tau$ be a topological on $‎\mathbb{N}‎$‎‎ such that is generated by $\{1,2\}, \{3,4\},\{5,6\}....$. Let $A$ be non-empty of $‎\mathbb{N}‎$‎‎ and $n_{0} \in A$. If $... 1answer 27 views ### A hereditarily Lindelöf$KC$-space$( X,τ )$is Katětov-$KC$if and only if there is a weaker sequential$US$topology$σ⊂τ

A space $( X,τ )$ is said to be Katětov $KC$ if there is a topology $σ⊂τ$ such that $( X,σ )$ is minimal $KC$. The notion of strongly KC-spaces, that is, those spaces in which every ...
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### An infinite minimal strongly KC-space possesses a non-trivial

The notion of strongly KC-spaces mean spaces in which every countably compact subset is closed. a space $(X,‎\tau‎ )$ is said to be minimal strongly KC if $(X,‎\tau‎ )$ is strongly KC but no ...
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### $KC$ spaces imply $US$ spaces , but vise versa is false.

In the $US$ space , each convergent sequence has unique limit. In the $KC$ space , every compact subset is closed. It easy to show that $KC$ spaces imply $US$ spaces. The ...
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### if $X$ is a countable, compact $T_{1}$ space and $A ‎\subseteq‎‎‎‎ X$ then either $A$ is compact or…

Theorem: if $X$ is a countable, compact $T_{1}$ space and $A ‎\subseteq‎‎‎‎ X$ then either $A$ is compact or there is a sequence in $A$ converging to point of $X- A$. proof: Suppose ...
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### The set of finite “variations” of an unconditionally convergent series is pre-compact

Proposition: If $\sum x_i$ is an unconditionally convergent series in a Banach space $X$, then $S=\{\sum_{i=1}^n \varepsilon_ix_i:n\in\mathbb N, \varepsilon_i=\pm1\}$ is pre-compact. Proof: 1) ...
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### The set of “variations” of an unconditionally convergent series is compact

Proposition: If $\sum x_i$ is an unconditionally convergent series in a Banach space $X$, then $S=\{\sum \varepsilon_ix_i:\varepsilon_i=\pm1\}$ is compact. Proof: 1) $\{-1,1\}^{\mathbb N}$ is ...
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### Nets and compactness in topological spaces.

I am reading Kelley’s book on general topology. There are a few statements on nets there (chapter 2), but the characterization of compact sets in the language of nets is not given. How should we prove ...
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### Unit ball in $C[0,1]$ not sequentially compact

This question is taken from Saxe K -Beginning Functional Analysis. Show that the closed unit ball in $C[0,1]$ is not compact by proving that it is not sequentially compact. (It's assumed that we ...
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### Compact maps problem in Lax

In Functional Analysis of Peter Lax there are the following exercise Show that if $\bf C$ is compact and $\{{\bf M}_n \}$ tends strongly to $\bf M$, then $\bf CM_n$ tends uniformly to $\bf CM$. ...
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### Example of compactness of the set of elements of a convergent sequence.

An example in a book I am reading on general topology demonstrates the concept of compactness by showing that the set $E = \{s_n : n = 0,1,2,3 \ldots \}$ is compact in some topological space S. The ...
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### Uniform convergence of continuous functions with Lipschitz limit

Let $K \subset \mathbb R^d$ be a compact. Let $\phi_{\varepsilon} \colon K \rightarrow \mathbb R$ be continuous and converge uniformly to $\phi$. Suppose further that $\phi$ is Lipschitz continuous. ...
Let $(X, \rho)$ be a complete metric space and let $S_1,...,S_N: X\rightarrow X$ be continous functions. A nonempty compact set $C$ is said to be an attractor for the family $\{S_1,...,S_n\}$ if ...