# Tagged Questions

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### Proof verification related to the discrete metric

Can someone please verify my proof? Let $X_1$ be a set and let $d_1$ be the discrete metric on $X_1$. (a) Prove that every subset of $(X_1, d_1)$ is open. (b) Prove that if $(X_2, d_2)$ ...
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### Prove that $[0,1]$ is not isometric to $[0,2]$.

Prove that $[0,1]$ is not isometric to $[0,2]$. Suppose there is an isometry $f:[0,1]\to[0,2]$. Since f is continuous and surjective, the only values for $f(0)$ and $f(1)$ are $f(0)=0$ and ...
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### Can someone criticise my incorrect proof about a set being open?

In the question I have to decide whether the set $S=\{(x,y)\in\mathbb{R}^2\;|\;x/y\leq 7\}$ is open, closed or neither. I attempted to prove it was closed but it turns out it is neither can someone ...
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### Cauchy inequality proof

I am studying cauchy inequality proof from notes I have from my class$$(\forall\vec{x},\vec{y}\in\mathbb{R}^n):|\sum_{i=1}^{n}x_iy_i|\le||\vec{x}||\cdot||\vec{y}||$$ We choose $\vec{x},\vec{y}$. And ...
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### About interior of the frontier (proof-checking)

Let $M$ be a metric space, and $A \subset M$ an open set. Show that $\stackrel{o}{\widehat{\partial A}} = \emptyset$. ($\stackrel{o}{\widehat{\partial A}}$ is the interior of the frontier) I ...
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### Show graph is closed if f is continuous

I need help figuring out how to complete this proof: Show that the graph of $f$ is closed if $f$ is continuous.
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### Proving a metric induces the product topology

Let $(M,d)$ and $(N,d')$ be metric spaces. Prove that the product topology is induced by the metric $d_1((x,y),(x',y')=d(x,x')+d(y,y')$ and ...
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### Prove that $\mathbb{R}^k$ is separable

I'd like to show that $\mathbb{R}^k$ is separable. (A metric space is called separable if it contains a countable dense subset.) Here's what I have and I'd like to confirm with everyone to see if ...
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### $E$ Closed iff $\partial E \subseteq E$

I'm having trouble verifying my proof, would appreciate some input on this one. Let $(X,d)$ be a metric space with $E\subset X$. Suppose $E$ is closed in $X$, which means that $E=\overline{E}$. By ...
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### Is this proof correct - simple proof about distance in normed linear space

Let $X$ be a normed linear space, and let $M$ be a closed subset of $X$. Show that any $x\in X\setminus M$ has non-zero distance from M. My proof: Assume there exists an $x\in X$ such that ...
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### Proof that compactness can be characterized by closed sets.

If anybody would be willing to check to see if this proof is correct I would really appreciate it. Prove that a metric space $(X,d)$ is compact if and only if for any family $(C_i)_{i \in I}$ of ...
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### Property of a metric in the space of all the sequences of real numbers

A few weeks ago I had this problem, the adjoint-teacher solved it on class, and I thought I understood, but now I'm rechecking and there are a few things that aren't clear for me. So we defined this ...
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### Proving that Union of a finite number of complete subsets of metric space $(X,d)$ is complete.

We have following 3 definitions. Definition: Suppose that $(X,d)$ is a metric space. A sequence $(\textbf{x}_n)_{n\in \mathbb{N}}$ of points in $X$ is said to be a Cauchy sequence, if, given any ...
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### Proof on if a set is discrete

I would like to know how well I answered the following proof: was it concise? Was it elaborate/rigorous? Did I use incorrect notation? I would also like to know if the set is a $T_{1}$ space, such ...
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### Is this proof that all metric spaces are Hausdorff spaces correct?

Let $x$ and $y$ be distinct points of a metric space $M$. Prove that there exist in $M$ disjoint open sets $U$ and $V$ with $x \in U$ and $y \in V$. Let $U$ and $V$ be open balls centered at $a$ and ...
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### Proof of the Banach Fixed Point Theorem

I am presenting to you my book's verion of the proof of the Fixed Point Theorem: $\{T^i x_0\}$ has been shown to be a cauchy sequence. As we have a complete metric space, this cauchy sequence has a ...
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### Is this proof that every convergent sequence is bounded correct?

I've tried the following proof that a convergent sequence is bounded but I'm not sure if it is correct or not. Let $(M,d)$ be a metric space and suppose $(x_k)$ is a sequence of points of $M$ that ...
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### Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous.

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous. ($M(n,\mathbb R)$ is identified with $\mathbb R^{n^2}$ as a normed liner space.) ...
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### Let $x=(x_n)$ be a Cauchy sequence in a metric space $X$ having a convergent subsequene

Let $x=(x_n)$ be a Cauchy sequence in a metric space $X$ having a convergent subsequene $x\sigma=(x_{\sigma_n})$ where $\sigma:\mathbb N\to\mathbb N$ is strictly increasing. Then $(x_n)$ is ...
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### Justify: In a metric space every bounded sequence has a convergent subsequence.

Justify: In a metric space every bounded sequence has a convergent subsequence. My Attempt: False: Consider the metric space $(X,d)$ where $X=\mathbb R$ and $d$ is the discrete metric on $X.$ ...
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### Properties for interior and closure in metric space.

I found the some following properties for general topology and prove these. But, I want to verify that the proofs are really true. Let $(X,d)$ be metric space. Let $A$ be any subset of $X$. Define ...
### Show that the diagonal $\{(x,x): x\in X\}$ is closed in the metric space $(X\times X,d=\max\{d_X,d_X\})?$
Show that the diagonal $\{(x,x): x\in X\}$ is closed in the metric space $(X\times X,d=\max\{d_X,d_X\})?$ My attempt: Choose $(x,y)\in X\times X-\{(a,a): a\in X\}$ Then $c=d(x,y)/2>0.$ To ...