0
votes
0answers
92 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 ...
0
votes
0answers
15 views

How to use a base to prove something is sequentially compact.

I know this is not very specific but I'm studying for a topology exam and this is one of the things I need to know how to do. I know that part of the process is showing it converges. I was hoping ...
2
votes
1answer
67 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
1answer
38 views

Strong convergence of bounded sequences in Bochner spaces

Let $S=(0,T)$ for a $T>0$ and let $B_0,\ B_1,\ B_2$ be Banach spaces, such that $B_0$ is compactly embedded in $B_1$, which is in turn continuously embedded in $B_2$. Suppose we have a sequence ...
0
votes
1answer
58 views

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, ...
2
votes
0answers
22 views

Compactness in topology of uniform conergence (of functions and all their derivatives) on compact subsets of (0,\infty)

I am trying to understand an example in the book "Lectures on Choquet's Theorem" (R.R. Phelps). My question is: Given the space of real valued infinitely differentiable functions on $(0, \infty)$ ...
2
votes
1answer
95 views

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 ...
1
vote
1answer
31 views

Show that $A$ is non-compact

I have a problem: For $C\left [ 0,1 \right ]=\left \{ x:\left [ 0,1 \right ] \to \Bbb R \ \text{is continuous on } \left [ 0,1 \right ] \right \}$, with a norm: $$\left \| x \right \|=\sup_{t\in ...
2
votes
1answer
38 views

Product with exponential converges absolutely and uniformly

Prove that the product $$\prod_{n=1}^\infty \left(1+\frac{z}{n}\right)e^{-z/n}$$ converges absolutely and uniformly on every compact set. What can I transform this product into? It's ...
1
vote
1answer
62 views

convergence of compact sets

Let $X$ and $Y$ be compact sets (both subsets of the real line). Assume $Y$ has a non-empty interior. Consider the continuous function $f:X \rightarrow Y$. For any given $y$ in the interior of Y, and ...
2
votes
1answer
47 views

Convergence of compact sets continuous function

Let $X$ and $Y$ be compact set (both subset of the real number). Consider the continuous function $f:X \rightarrow Y$. For any given $y$, and for $h>0$ small enough so that $y+h \in Y$. I want to ...
0
votes
1answer
155 views

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. ...
3
votes
1answer
82 views

Need a unique convergence (UC) space's Alexandrov extension be a UC space?

Background Say a topological space $X$ is a unique convergence (UC) space iff every sequence of points of $X$ converges to at most one point of $X$; a unique convergent clustering (UCC) space iff ...
2
votes
1answer
159 views

Totally bounded subset in complete metric space implies compact?

I am reading the book Elements of Functional analysis by Kolmogorov and Fomin. In chapter 2, section 16 on compact metric spaces the author poses the following theorem which he demonstrates ...
0
votes
1answer
75 views

Compact spaces where not all compact subsets are closed

A topological space $(X,\tau)$ is called $C-C$ iff the closed sets in $X$ coincide with the compact sets in $X$. A topological space is called a $US$-space provided that each convergent sequence has ...
2
votes
1answer
49 views

Is a locally compact space a KC-space if and only if it is Hausdorff?

A topological space is called a $US$-space provided that each convergent sequence has a unique limit. We know that for locally compact spaces,‎ ‎$ ‎T‎_{2} ‎‎‎\equiv KC‎‎$. We have: ‎‎$ ‎T_2‎ ...
1
vote
0answers
43 views

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 ...
0
votes
0answers
88 views

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 ...
1
vote
1answer
44 views

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 ...
1
vote
1answer
20 views

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. ...
1
vote
1answer
48 views

Accumulation point

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 $ ...
1
vote
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 ...
1
vote
1answer
31 views

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 ...
0
votes
2answers
82 views

$ 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 ...
1
vote
1answer
79 views

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 ...
1
vote
1answer
47 views

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) ...
1
vote
1answer
31 views

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 ...
6
votes
2answers
214 views

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 ...
5
votes
2answers
637 views

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 ...
8
votes
2answers
222 views

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 $\bf CM$. ...
1
vote
0answers
201 views

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 ...
1
vote
1answer
279 views

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. ...
0
votes
0answers
40 views

The existence of an attractor for a finite family of continuous transformations

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 ...
8
votes
1answer
429 views

Stone-Čech compactifications and limits of sequences

I've been working on some old prelims from my university when they used to just be on point-set topology. We don't cover a couple of the topics so I've been teaching myself some of the material, one ...