# Tagged Questions

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### 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 $x$.
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### 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 $B$ ?
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### 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 ...
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### 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 ...
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### Topology Hausdorff $\Leftrightarrow$ Unique Limits [closed]

Prove that a topology is Hausdorff iff every net (not filter!) has at most one limit.
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### 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 ...
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### 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 $0.$ ?
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### 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 $f, g \in X$, ...
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### 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 ...
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### 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. ...
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### 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 ...
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### Cauchy filters defined for proximity spaces?

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