0
votes
1answer
25 views

In the semi linear uniform space

In the semi linear uniform space, If $f$ is a function from $(X ,Γ_X)$ to ($Y,Γ_Y)$ where $f(x_n)$ converges to $f(x)$ whenever $x_n$ converges to $x$,show that $f$ is continuous at $x$.
0
votes
1answer
56 views

Density and convergence

I have a small question: Is it true that if the basis of a space $A$ is dense in a space $B$ ($B\subset A$) then if $u_n\rightarrow u$ in $A$ we have that $u_n\rightarrow u$ in $B$ ?
1
vote
2answers
60 views

sequential continuity vs. continuity

A short and hopefully simple question for someone with more experience in topology: If a topology is induced by a mode of convergence and in fact nothing more is known apriori, wether if this topology ...
2
votes
1answer
38 views

Relations between convergence in nets and topologies.

I want to prove that given a net $S$ and topologies $T$ and $T'$, then $T\subset T'\iff$ when $S$ is convergent in $T'$ is also convergent in $T$. I'm proceeding this way: First I'd like to show that ...
0
votes
1answer
87 views

Topology Hausdorff $\Leftrightarrow$ Unique Limits [closed]

Prove that a topology is Hausdorff iff every net (not filter!) has at most one limit.
4
votes
1answer
101 views

In search for a topology

I'm looking for a way to convergence on subspaces. If $G_k(\mathbb{R}^m)=\{ W: W$ is subspace of $\mathbb{R}^m, \dim W=k \}$ and consider in $G_k(\mathbb{R}^m)$ one topology $\tau$. I would like to ...
1
vote
1answer
35 views

Existence of net of positive real numbers

Does for every totally ordered infinite set $T$ with no greatest element, there exist a net $(a_t )_{t\in T}$ of positive real numbers which is convergent to $0.$ ?
0
votes
1answer
62 views

Show that a metric space is complete

I have the following problem: Let $X$ denote the collection of all differentiable continuous functions $f : [0, 1] \rightarrow \Bbb R$ such that $f(0) = 0$ and $f'$ is continuous. For $f, g \in X$, ...
6
votes
1answer
117 views

These germs make me sick!

I need a "mini-crashcours" concerning the space of germs of continuous functions in order to solve an exercise which requires me to show that limits in this space aren't always unique. We have ...
1
vote
1answer
51 views

Subspaces and convergence in weak* topology

I would like to ask some questions regarding convergence in the weak* topology and subspaces. Let $X$ be a normed space with subspace $A \subset X$. Assume $X$ is endowed with the weak* topology. ...
0
votes
0answers
97 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
75 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 ...
3
votes
1answer
47 views

How much close should two spaces be in the Gromov-Hausdorff distance to be homeomorphic?

Here I am considering the Gromov-Hausdorff convergence for metric spaces. I know two compact metric spaces are isometric if and only if $d_{GH}(X,Y)=0$, where $d_{GH}$ denotes the Gromov-Hausdorff ...
2
votes
1answer
33 views

Conditions for convergences of a net

I got stuck on this problem and got no clue to solve it. Can anyone one here help me? I really appreciate. Let $X$ be a set and $\mathcal{A}$ the collection of all finite subsets of $X$, ...
0
votes
1answer
39 views

Topology of a convergence space

I am actually having an introduction to filters. Today I was trying to prove that the collection of open sets of a convergence space satisfy the axioms of a topology: O $\subset$ X is open iff $lim ...
0
votes
1answer
26 views

Opens in a convergence space

By the book "contemporary mathematics", Beyond Topology (F.mynard , E.Pearl) I am now studying convergence spaces on the book mentioned above. On this book (p.123) I find this definition: A subset O ...
0
votes
1answer
33 views

Cauchy filters defined for proximity spaces?

I define in my draft article Cauchy filters $\mathcal{X}$ on a uniform space $\nu$ by the formula: $$\mathcal{X}\ne\bot \wedge \mathcal{X}\times^{\mathsf{RLD}}\mathcal{X}\sqsubseteq\nu.$$ ...
0
votes
1answer
82 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, ...
0
votes
1answer
79 views

Do the supremum and infimum always exist for convergent sequences

If $\{x_n\}$ is a real sequence converging to $x \in \mathbb{R}$, do $\displaystyle \sup_n{x_n}$ and $\displaystyle\inf_n{x_n}$ exist? I think yes as, choosing $\epsilon = 1$ we have for some ...
1
vote
1answer
68 views

Showing that a regulated function belongs to Schwartz Space

For clarity I am using the following definition of Schwartz space $\mathscr{S}$ Let $\mathscr{S}$ be the set of functions f(x) s.t. the family of norms ...
1
vote
1answer
64 views

What are the typical approaches to showing that some function sequence does not converge uniformly?

The following problem is from Munkres's Topology (Exercise 6 of Section 21 "The Metric Topology (continued)", 2nd edition). Exercise: Define $f_n : [0,1] \to \mathbb{R}$ by the equation $f_n(x) = ...
3
votes
2answers
66 views

Convergent sequence in product space on $\mathbb{R}^{\omega}$

I am confused about the concept of convergent sequence in product space when learning Munkres's Topology, especially when I am comparing two related exercises of it. The exercise 6 of section 19 ...
2
votes
0answers
26 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 $(0, \infty)$ ...
4
votes
1answer
60 views

Pointwise convergence to a constant function on a compact space

I would like to show that there is no sequence of homeomorphisms of a compact metric space which converges pointwise to some constant function. However, I'm not sure this result is true; in the case ...
0
votes
1answer
49 views

Countable sum of closed boundary sets

I have to prove that for complete metric space and $f_n$ converge pointwisely to $f$ $f^{-1}(a,b)\setminus Int(f^{-1} (a,b)) $ is countable sum of closed, boundary sets. Here is my solution: ...
6
votes
1answer
344 views

Equicontinuity implies uniform convergence

So I know it's a theorem that if $\{f_n\}$ is a sequence in an equicontinuous family of functions defined on a compact metric space $K$ then if for all $x$, $f_n(x)\rightarrow f(x)$ pointwise then ...
2
votes
0answers
81 views

Pointwise convergence on a complete metric space

Let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
2
votes
0answers
76 views

Discontinuous function on Q

let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
2
votes
2answers
47 views

how to prove that the limit of this sequence of functions is continuous?

I have a norm that works in function space and that is $‖∙‖_{sup}:C([0,1])→R$, $‖∙‖_{sup}:=sup${$|f(t)|$}. I need to show that the metric is complete. So I need to show that every Cauchy sequence of ...
2
votes
2answers
56 views

On eventually constant sequences

It is of course true that in a discrete space a sequence converges iff it's eventually constant. Is the converse true, i.e., if the only convergent sequences in a space are eventually constant, is the ...
2
votes
1answer
75 views

There exists function sequence $\{f_{n}\}$ converges to $0$ such that $\{a_{n}f_{n}\}$ not converges to $0$

Let $X$ be the vector space of all complex functions on the unit interval $[0,1]$, topologized by the family of seminorms $$p_{x}(f) = |f(x)|, \quad (0 \le x \le 1).$$ Show that there exists a ...
1
vote
1answer
39 views

Question about strong and norm convergence.

Maybe the answer to this question is so trivial that I can't see it: Why the strong convergence of operators (on an hilbert space) does not imply the norm convergence? Many books make this example: ...
0
votes
3answers
119 views

Help with Pointwise and Uniform Convergence in Metric Spaces

I am having a bit of difficulty understanding uniform convergence and would also like to check my understanding of pointwise convergence. Using the example of $f_n$(x) = $x^n$ on (-1,1), I found the ...
2
votes
1answer
33 views

Does $T_1$ imply Fréchet–Urysohn (every limit point is a limit of some sequence)?

It's not a problem from a book so I’m not even sure the statement is true. Nevertheless here's an alleged proof: ADDED LATER: Although the result is wrong I can't find a problem with the proof. I ...
0
votes
2answers
59 views

the power series converges in compact convergence topology

Consider the sequence of functions $f_{n}: (-1,1) \rightarrow R$ defined by:$$f_{n}(x) = \sum_{k=1}^{n}{kx^{k}}$$ a) Prove that $(f_{n})$ converges in the topology of compact convergence, ...
0
votes
1answer
50 views

Is the topological space $\mathbb{R}$ sequential?

A topological space is called a sequential space if a set $A ⊂ X$ is closed if and only if together with any sequence it contains all its limits. A topological space is called a Frechet space, if for ...
0
votes
1answer
92 views

Convergence of sequence in uniform and box topologies

I am trying the following problem: $w_1=(1,1,1,1,\ldots)$ $w_2=(0,2,2,\ldots)$ $w_3=(0,0,3,3,\ldots)$ $\cdots$ $x_1=(1,1,1,1,\ldots)$ $x_2=(0,\frac{1}{2},\frac{1}{2},\frac{1}{2}\ldots)$ ...
0
votes
1answer
25 views

If $x \in \overline{A}$ and $A \subset X$, $X$ first-countable, then there's a sequence of points in $A$ converging to $x$.

Supposedly this relies on first-countability of $X$. Let $x \in \overline{A}$, then by definition there's a neighborhood $U_1$ of $x$ that contains some $x_1 \in A$. If this is the only ...
4
votes
1answer
116 views

topology homework

I'm new to topology, and therefore not very good at it yet. I have following questions, that I have ansewer, please help me verify what is not correct and what is missing in my answers. Let $X$ be ...
3
votes
1answer
93 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 ...
0
votes
1answer
49 views

topological sequential space $(X,\tau)$

Suppose in topological space $(X,\tau)$ every countably compact is closed.Let $(X,\tau)$ be sequential space. (1): If every infinite subset $A \subseteq X$ is closed, will $A$ be discreet in $X$? ...
4
votes
2answers
235 views

Bounded derivative implies bounded function?

By the following theorem, it suffices to show that $\{F_n: n\in\mathbb N\}$ is equicontinuous and bounded: If $f_k$ is a sequence in an equicontinuous and pointwise bounded set of maps from a ...
0
votes
1answer
131 views

Is a set of bounded functions bounded?

Please consider the following question (note that $C_b$ is the space of bounded continuous functions): Let $f_k$ be a convergent sequence in $\mathscr C_b(A, \mathbb R^m)$. Prove $\{f_k \mid k = ...
0
votes
1answer
80 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 ...
4
votes
1answer
101 views

Fréchet-Urysohn space

We know that if $p$ is the limit of a subsequence $(x_{n_{k}})$ of the sequence $(x_n)$ in $X$ then $p$ is a cluster point of the sequence. For sequential spaces it does not to hold that a cluster ...
3
votes
2answers
479 views

Convergence in a Topological Space

I am trying to figure out how to prove the following problem from my topology homework: "Let X have the discrete topology and let an -> b. Prove that the sequence must be eventually constant; that ...
1
vote
1answer
40 views

A topological space is called a US-space provided that

A topological space is called a US-space provided that each convergent sequence has a unique limit. Each Fréchet $US$-space $X$ is a $KC$-space. Proof. Suppose that $x \in K$ where $K$ is a ...
-1
votes
1answer
62 views

Let $X$ be any topological soace which satisfies the first axiom of countability. If every compact subset of $X$ is closed, then $X$ is Hausdorff.

Let $X$ be any topological soace which satisfies the first axiom of countability. If every compact subset of $X$ is closed, then $X$ is Hausdorff. proof: Suppose that every compact subset of $X$ ...
1
vote
2answers
144 views

Cluster Point in Fréchet-Urysohn Space is Limit Point?

A topological space is called a Fréchet-Urysohn space if for every $A \subset X$ and every $x \in \overline{A}$ there exists a sequence $x_1, x_2,....$ of points of $A$ converging to $x$. So, is it ...