Tagged Questions

1
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2answers
88 views

Continuous functions uniformly convergent to a function, metric spaces, equivalent conditions

Let $X, \ (Y, d)$ be metric spaces, $f_1, f_2, \ldots \ : X \rightarrow Y$ be continuous functions, $f: X \rightarrow Y$ an arbitrary function. Prove that the following condtions are equivalent: 1) ...
3
votes
1answer
31 views

Baire one functions, closed intervals

I've been wondering if you could help me with the following problem. There's an article on Baire one functions (number 2 on google search list) and there is one thing concerning Lemma 9 that I am not ...
0
votes
1answer
58 views

Open or closed set of converging sequence

I have come across a question regarding whether a set is open or closed... Needing to give details of why its open or closed.. $C_0 $ the set of sequences converging to $0$ in $(\ell^{\infty} ...
1
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5answers
150 views

Can a sequence of finite subsets of $\mathbb R$ converge to $\mathbb R$?

Assume we have a sequence of finite subsets of $\mathbb R$. Or equivalently a sequence of finite subsets of the closed interval from $0$ to $1$. Is it possible that this sequence converges to the ...
1
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1answer
42 views

Show that $\displaystyle \sum_{n\geq{1}} ||x_n -x_{n+1}|| < \infty$ is a Cauchy sequence. [duplicate]

If a sequence $(x_n)^{\infty}_{n=1}$ in $\mathbb{R}^n$ satisfies $\displaystyle \sum_{n\geq{1}} ||x_n -x_{n+1}|| < \infty$, Show that it is a Cauchy sequence. Thoughts: By definition, a ...
5
votes
2answers
120 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 ...
2
votes
1answer
69 views

Moore–Smith convergence; Universal nets

I am looking at John Kelley's General Topology. This is exercise J from chapter $2$. I seem to be having trouble with problems c) and d). c) Prove the following Lemma: If $S$ is a net in $X$, ...
1
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2answers
169 views

prove Taylor of $R(a)$ converges $R$ but its sum equals $R(a)$ for $a$ in interval.Which interval? Pls I'm glad to give an idea or hint?:

$T(a)=\begin{cases} 0 & a\leq 0\\ e^{-1/a} & a>0 \end{cases}$ $Z(a)=\begin{cases}0 & a\geq 1\\ (1+a) e^{-1/(1-a)} & a<1 \end{cases}$ $p=\int_{-1}^{1}Z(x)dx$ ...
0
votes
1answer
33 views

Convergence properties of nets and filterbases

Suppose $\mathcal{B}$ is a filterbase in $X$. If for each $B \in \mathcal{B}$, we have $x_B \in X$, then $\lambda: \mathcal{B} \rightarrow X$ such that $B \mapsto x_B$ is a net in $X$. Show that ...
1
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2answers
40 views

Net Convergence

Let $\Gamma$ be a uncountable set and $\psi(\Gamma)\subset\mathbb{R}^{\Gamma}$ be the space of all countably supported elements. How can I show that $\overline{\psi(\Gamma)}=\mathbb{R}^\Gamma$, i.e. ...
4
votes
1answer
48 views

If $\mathcal{B}$ in $X$ converges to $x$, it accumulates at $x$ and if $X$ is Hausdorff, $x$ is a unique point of accumulation.

Essentially I need feedback, mostly on writing style and accuracy and tightness of arguments. Suppose $\mathcal{B}$ converges to $x$. For every open neighbourhood $U(x)$ of $x$ in $X$ there exists ...
3
votes
1answer
54 views

Hausdorff iff convergent filterbase has a unique point of convergence. Is this correct?

Suppose $(X, \mathcal{T})$ is a Hausdorff space. Consider a filterbase $\mathcal{B}$ in $X$ such that $\mathcal{B}$ converges to $a\in X$.We show that, if $b \in X$ and $a\neq b$, then $\mathcal{B}$ ...
2
votes
2answers
45 views

Hausdorff spaces and convergence

How can I construct a filterbase converging to two points in in a non-Hausdorff space?
2
votes
2answers
63 views

Nets and filterbases and convergence

I am not sure I understand the difference between nets and filterbases very clearly, esp since there convergence properties seem to coincide. Isn't a filterbase a kind of a net?
2
votes
1answer
61 views

Subspace $Y$ of metric space with finitely many points is complete.

Show that if a subspace $Y$ of a metric space consists of finitely many points, then $Y$ is complete. This is what I have so far, but I don't know where to go from here: Suppose the the subspace ...
2
votes
1answer
55 views

Convex functions and uniform convergence of derivatives

Let $f_n:[0,1]\to\mathbb{R},\ n\in\mathbb{N}$ be a sequence of convex analytic functions. Consider the sequence of derivatives $(f_n')_{n\in\mathbb{N}}$. Suppose that ...
6
votes
2answers
78 views

Closeness of probability measures

Consider a set of probability measures $\{P_n\}$. Suppose $P_n$ converges to $P^*$ weakly and $$ \int \xi^2 P_n(d\xi)< \infty. $$ Can we claim $$ \int \xi^2 P^*(d\xi)<\infty $$ and $$ \lim_{n\to ...
7
votes
1answer
77 views

Convergence in the Box Topology.

Given the sequence in $\mathbb{R}^\omega$: $$y_1=(1,0,0,0\ldots), y_2=(\frac{1}{2},\frac{1}{2},0,0\ldots), y_3=(\frac{1}{3},\frac{1}{3},\frac{1}{3},0,\ldots),\ldots$$ I know that it converges in the ...
4
votes
1answer
117 views

convergence of nets vs. convergence of really long sequences

It is well known that for a function $f:X\to Y$ between the underlying sets of topological spaces, the condition that $f$ is continuous is equivalent to the condition that given any net $N$ in $X$ ...
1
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1answer
161 views

Weak closure of $\{\sqrt n e_n|n\in \mathbb N\}$ and metrizability of weak topology

Let $\{e_n|n\in \mathbb N\}$ be an orthonormal basis of Hilbert space $\mathcal H$ and put $I = \left\{\sqrt n e_n|n\in \mathbb N\right \}$. Show that $0$ belongs to the weak closure of I but no ...
2
votes
2answers
75 views

Convergence in $\ell^1$

Could anyone help with the proof of the following: the standard basis of $l_1(\mathbb{N})$ does not have a limit in weak topology? I think it is the case in norm topology since that sequence is not a ...
4
votes
1answer
114 views

Prove that C[X] with the pointwise convergence topology is not metrizable

I need to show that if X is an uncountable Tychonoff space, then C[X] is not metrizable. All I've been able to show so far is that that F[X], the space of all functions with pointwise topology, is ...
1
vote
1answer
90 views

Uniform convergence of functions, Spring 2002

The question I have in mind is (see here, page 60, the solution is at page 297): Assume $f_{n}$ is a sequence of functions from a metric space $X$ to $Y$. Suppose $f_{n}\rightarrow f$ uniformly and ...
6
votes
1answer
93 views

Sequences, subsequences, and continuity of functions

It's been a few years since I studied point-set topology, and I'm a bit rusty on the basics. Would appreciate help with the following question. Suppose $f:X\rightarrow Y$ is a map between two ...
2
votes
2answers
78 views

Is the space of continuous function with given order of decrease closed?

For example, denote $O^1$ the space of continuous function with property $\lim_{x\to\infty}{|x|f(x)}=0$ or $f(x)=o\left(|x|^{-1}\right)$ as $x\to\infty$. It's obviously an intermediate vector space ...
0
votes
1answer
57 views

How to compatilize convergence of functions as points and pointwise convergence of functions?

In general, let $X$ be a set, $\mathcal N:X \to \wp(\wp(X))$ assign neighbourhood system to every point, $B$ be a filter basis on $\wp(X)$, then we say $B$ converges to $x$, or $B \to x$ iff ...
2
votes
1answer
185 views

Limsup of continuous functions between metric spaces

Let me start with a simple example: Let $f_n:[0,1]\to[-1,1],x\mapsto \sin 2\pi nx$. For each $x\in[0,1]$, consider the sequence $\lbrace f_n(x):n\ge1\rbrace$ and denote by $F(x)$ the set of points of ...
2
votes
1answer
66 views

Union of compact sets in a convergence space

Let $X$ be a convergence space and let $K_1, K_2, \ldots, K_n$ be compact subsets of $X$. I'm trying to prove for myself that the union $K$ of the $K_i$ is compact. By definition, $K$ is compact if ...
2
votes
1answer
102 views

Weak a.s. convergence VS a.s.weak convergence

Let's consider a sequence $(\mu_n)_n$ of random probability measures on $\mathbb R$, and let $C_b$ be the Banach space of bounded continuous functions on $\mathbb R$. I am considering the following ...
1
vote
2answers
70 views

Verifying Properties of Filters

I'm on chapter 5 now in these notes: http://math.uga.edu/~pete/convergence.pdf I'm stuck trying to prove Proposition 5.6. (on the top of page 21/bottom of page 20). First, I think that the first ...
2
votes
3answers
168 views

In a first-countable space, an accumulation point of the set of terms in a sequence is also a limit-point of the sequence

I ran into something which seems like it should be true (and must be true because it is used in the proof of a theorem). I've extracted the detail that I cannot quite verify. Let $(x_{n})$ be a ...
3
votes
1answer
93 views

Proof of Kelley's Theorem

So, still working through the convergence notes here: http://math.uga.edu/~pete/convergence.pdf And I was wondering if somebody could help me finish the proof of Kelley's Theorem, which states that ...
1
vote
2answers
74 views

weak convergence on $\ell_1$

Let $\{x^k\}$ be a weak convergent sequence in $\ell_1$, and its weak limit is 0. Is the following property true: For $\forall \epsilon >0$ and $\forall n>0$, there exists a K, s.t. ...
4
votes
1answer
172 views

Convergence without metric or topology or sigma field.

You can set some kind of convergence in a space of functions without using some metric or topology or sigma field?
1
vote
1answer
145 views

Relationship between Convergence and Open sets

If you show that convergence of nets in a topological vector space $V$ with topology $\tau$ is equivalent to convergence of nets in a topological vector space $V$ With topology $\sigma$, does it ...
3
votes
1answer
54 views

Disjointly supported functions

Let $(f_n)_n$ be sequence of real-valued continuous functions on a compact, Hausdorff space $K$ with pairwise disjoint (closed) supports satisfying $$0<\inf_n \|f_n\|\leq \sup_n\|f_n\|<\infty.$$ ...
3
votes
1answer
239 views

Pointwise limit of continuous functions

Given a compact Hausdorff (I do not assume metrisability) space $K$ and a sequence $(f_n)_n$ of continuous real-valued functions on $K$ such that the pointwise limit of this sequence exists. Must the ...
4
votes
4answers
301 views

Does the completeness of a normed vector space only depend on its topology?

Let $V \space$ be a vector space over $\mathbb{R}$, and $\Vert \cdot \Vert_1$, $\Vert \cdot \Vert_2$ norms over $V$, which generate the same topology. Is it always true that if $v_n$ is a Cauchy ...
6
votes
1answer
662 views

Does the p-norm converge to the max-norm in some norm

Does the $p$-norm on $\mathbf{R}^n$ converge to the max-norm on $\mathbf{R}^n$ as elements in the space of real valued continuous functions on $\mathbf{R}^n$ endowed with some norm? More precisely, ...
9
votes
1answer
189 views

Convergence in topologies

Let $\tau_1\subseteq \tau_2$ be two Hausdorff regular topologies in an infinite set $X$ such that the convergences of sequences in $\tau_1$ and $\tau_2$ coincide (they have the same convergent ...
8
votes
5answers
825 views

Is there a way of working with the Zariski topology in terms of convergence/limits?

As someone who is very fond of analysis, I feel most comfortable working in topological spaces via the notion of convergence of sequences (or nets, in infinite-dimensional Banach spaces, etc.). In ...