Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Calculate X Y Z from two specific degrees on a sphere

I am a programmer, don't know much about advanced math. I would need the exact formula(s) that could achieve this, so I can translate it to my programming language. I am having a headache trying to ...
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21 views

Defining metrics as a function to something other than the reals.

Generally speaking, a metric for a space R is defined as a function from RxR -> Reals, but does it have to be? Can we define it in more generic terms such as a function from R to a field with certain ...
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25 views

Is $A$ compact, $f(A)$ uniformly continuous and is $f^{-1}$ continuous?

$X$ and $Y$ are metric spaces, $A\subseteq X$, $A$ is bounded. map $f:X\to Y$ is continuous. Questions: Is $A$ necessarily compact? Is $f(A)$ uniformly continuous? If given that $f$ is a ...
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How do you prove that Z (the set of integers) equipped with the Euclidean metric (induced from real numbers) is complete?

How do you prove that $\mathbb{Z}$ (the set of integers) equipped with the Euclidean metric (induced from real numbers) is complete? I am having trouble with this question, I don't really know where ...
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23 views

On Equivalent Norms in an Infinite Dimensional Vector Space

How many non-equivalent norms can we define in an infinite dimensional vector space? Is there any explicit expression?
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How to define distance between two functions in a non-linear space (example of non-linear space: shape space)?

Suppose I have two parametric circle $f_1=(acost,asint)$ and $f_2=(bcos t,bsint)$, $t\in(0,2\pi),a>0,b>0$, which lies in some non-linear space. Are there any way, how to define the ...
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25 views

When a metric space is a normed space?

I'm trying to figure out that which condition should be provided for a metric space to be normed also?
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If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable

I need to show that: If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable. I have already showed that every locally compact Hausdorff space ...
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Prove the set, {y ∈ X | r ≤ d(x,y) ≤ s}, is closed

Let r < s be positive real numbers and x ∈ X. Prove that the set: {y ∈ X | r ≤ d(x,y) ≤ s}, is closed. Having trouble with how I should tackle this ...
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45 views

Retraction to an interval in a metric space

Suppose that $X$ is a metric space and $A$ is a subspace of $X$ that is homeomorphic to the interval $[0,1]$ with its usual topology. Let $v$ and end point of A. How do you proof that there is a ...
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13 views

A Base of a metric space intuition

From what I have read online and from what I have read in Rudin, a collection of open sets $\lbrace$$V_{n}$$\rbrace$ is said to be a base for a metric space $X$ if every open set in $X$ can be ...
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28 views

Cantor's intersection Theorem without the diameter hypothesis

In proving Cantor's in intersection theorem, the fact that limit of the diameter of the sets is 0 was used to prove that the intersection is non-empty. I just wondered if that hypothesis is excluded ...
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19 views

Calculate projection of a line in a square

Said that we have two points (P1, P2) that form a line, and 3 points (S1,S2,S3) that form a square, how would we calculate the position X and Y of the point resulting from the intersection of the line ...
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19 views

Topology of metric completion of Euclidean metric

Lets consider $\cal{M}=\mathbb{R}^{2}\backslash\{(0,y)\}\text { with } \{|y|\le1\}$ with the Euclidean metric with line element $ds^{2}=dx^{2}+dy^{2}$. Now consider the distance function given by ...
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Problem about completeness

Does there exist a complete metric on $(0,1)$ inducing the usual topology? My problem is that I cant understand what will I have to do to answer the question.It's a problem of a competitive exam.
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Showing the right half of the unit hyperbola is a complete metric space.

Let $f : \mathbb{R} \rightarrow \mathbb{R}^2$ be given as follows. $$f(\theta) = (\cosh \theta, \sinh \theta)$$ I want to argue that $\mathrm{im}(f)$ is a complete metric space with respect to the ...
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Normed Space and Hibert Space Problem

Anyone could describe me, why this is True? Suppose $(H, \|.\|) $ is a normed space. the norm $\|.\|$ induced by an inner product if and only if Parallelogram law is valid. Regards.
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Example of $x$ being adherent point but not accumulation point?

So I was just reading Apostol and I see that if $x$ is an accumulation point of set $S$, it has to be an adherent point as well. I guess it's possible for $x$ to be an adherent point only, not an ...
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Let $(X,d)$ be a metric space, $C$ a compact subset, and $K$ a closed subset. Prove that $K \cap C = \emptyset$ iff $d(K,C) > 0$.

Let $(X,d)$ be a metric space, $C$ a compact subset, and $K$ a closed subset. Prove that $K \cap C = \emptyset$ iff $d(K,C) > 0$. For this problem I was going to consider $d(x,F) = \inf d(x,y)$ ...
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47 views

How to show that addition is continuous?

Let $f: R \times R \rightarrow R$ and let the metric over $R$ be $d(x,y)=|x-y|$ and let the metric in $R \times R$ be $d_2((x,y),(a,b))= ((x-y)^2+(a-b)^2)^{1/2}$. I believe I understand how to ...
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Is every connected subset of the Sierpiński triangle arcwise connected?

I think this should be true. If it's indeed the case, it seems like this should be a known result, so references are welcome. I managed to prove that (assuming $S$ is the connected subset) $S$ ...
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Convergence of functions in a metric space

Let $C([0,1])$ be the space of all continuous functions from $[0,1]$ to $\mathbb{R}$ under the metric $$ \lVert f \rVert_1 = \int_0^1 \lvert f(x) \rvert \, dx. $$ Now consider $f_n(x) = e^{-nx}$. I ...
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Proving a set is compact - Homework

Let $(X,d)$ be a metric space and let {$p_n$} be a sequence of points in $X$ with $\lim_{n\to ∞}p_n = p_0$. Prove that the set $K =$ {$p_0, p_1, p_2,...$} is a compact subset of $X$. I have ...
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Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true?

Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true for all $x,y\in\mathbb{R}$? If not, how can I prove that $\int\frac{\vert ...
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17 views

Closed Interval in $E^2$

I am currently working through Introduction to Analysis by Rosenlicht In one of the exercises $4.30,$ he asks a question regarding a closed interval in $E^2.$ I am not sure what this means. I was ...
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30 views

Any open set shares boundary with a discrete set

Claim: Let $X$ be a metric space and let $U\subset X$ be open. Then there exists a discrete set $A\subset X$ such that $\partial A = \partial U$. Approach thus far: Since this statement is about ...
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Problem in proving norm

I have a linear space V that includes the continuous functions from [-$\pi, \pi$] to the complex set C, of the form: $$f(x) = a\cos(t)+b\sin(t) $$ where $a$ and $b$ are complex numbers. I want to ...
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59 views

Find sets of points, where function from one topological space to another is continuous.

We have got two functions : $f(x,y) = (2x,y)$ $g(x,y) = (x+1,y) $ They are transormations from one topological space to another ( from $ (\mathbb{R^2}, \tau')$ to $ (\mathbb{R^2}, \tau'')$ ), ...
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Necessity in Arzela-Ascoli theorem

I am trying to prove necessity of boundedness and equicontinuity in Arzela-Ascoli and I don't know how to go about it. More precisely,I have: Let $K$ be a compact metric space, and $A\subset C^0(K)$ ...
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Having difficulties showing the triangle inequality of metric in the plane

Let $P \in \mathbb{R}^2$ and define $$ d(x,y) = \left\{ \begin{array}{lr} ||x-y|| & \text{if} \; \; x,y,P \; \; \text{are collinear,}\\ || x - P|| + ||y-P||& \;\;\;\; ...
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topology induced by euclidean metric is equivalent to manhattan metric in R?

How can we prove that the topology induced by euclidean metric is equivalent to manhattan metric in R?
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Distance function is in fact a metric

I know I should be able to show this, but for some reason I am having trouble. I need to show that $$d(x,y) = \frac{|x-y|}{1+|x-y|}$$ is a metric on $\Bbb R$ where $|*|$ is the absolute value metric. ...
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How to compute uniformly distributed points on an ellipse

The ellipse can be parametrized in polar coordinates by $$r(\theta)=\frac{1}{a+\cos\theta}$$ up to a scaling factor, and $a>1$. Suppose we measure $S$, the distance along the ellipse from the ...
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Disjoint open sets in $\mathbb{R}^N$

I have a set exercise which says: Prove that in $\mathbb{R}^N$ with the Euclidean metric any collection of disjoint open sets is at most countable. Is this true for any arbitrary metric space? Now I ...
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Continuous function with infimum

Let $A$ be a closed subset of a metric space $X$ and $f:A \rightarrow[1,2]$ a continuous function on it. Now I want to find out why the function $$F(t):=\frac{\inf\{f(s)d(s,t);s \in A\}}{\inf ...
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Closed sets in a subspace are formed by intersecting the subspace with closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
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1answer
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Distance to a set

I have a question concerning to the following problem. Let $(X,d)$ be a metric set. For every subset $T \subset X$ we define a mapping \begin{equation} d_T : X \rightarrow R , d_T(x) := inf\{d(x,y) | ...
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Existence of maximizer implies compact? [duplicate]

I know that compact sets imply the existence of a maximizer, but is the converse true: Let $(X,d)$ be a metric space. Suppose that whenever $f$ is a continuous (and real) function on $X$, there ...
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1answer
42 views

MAth proof questions Open closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
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1answer
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properties of connected set

If $C $ is a connected set in a metric space $X$ & $C$ intersects both $A$ and $X\cap A^c (A\subseteq X)$ then can it be concluded that $C\cap \delta A\neq\phi$ ?
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graph of connected set [closed]

Let $X,Y$ be connected metric spaces.Assume that $X$ is connected & that $f:X\rightarrow Y$ is continuous.Is $\{(x,y)\in X\times Y:y=f(x),x\in X\}$ connected in $X\times Y$(w.r.t. product metric)? ...
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Is $SO(2,\mathbb R)$ connected

Is $SO(2,\mathbb R)=\{ A\in O(2,\mathbb R): det A=1\}$ connected?Why?
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Topology-Open Sets of a Metric Space

Let $(X_i,d_i), i=1,2,\dots,n$ be metric spaces. Let $X=\prod_{i=1}^{n}X_i$ and let $(X,d)$ be the metric space defined in the standard manner. For $i=1,2,\dots,n$, let $O_i$ be an open subset of ...
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Question about Compact metric space

Please why $$\Delta_p=\lbrace (t_0,...,t_p)\in \mathbb{R}^{p+1},t_i\geq 0, \sum_{i=0}^p t_i=1\rbrace$$ is a compact metric space ? Thank you
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showing completeness of a metric space

$X=\mathbb{R}_{>0}$, $d(a_1,a_2)=|\ln(a_1)-\ln(a_2)|$. I have already proven that $(X,d)$ is a metric space, but I have some problems showing the completeness. Let $(a_n)_{n\in\mathbb{N}}$ a ...
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What is the maximal size of an equal-distance set in $\mathbb{R}^n$?

Let $A\subseteq \mathbb{R}^n$ with the casual metric and $c\in\mathbb{R}^+$ be a real positive number, such that for every $a_1, a_2\in A$ if $a_1\neq a_2$ then $d(a_1,a_2)=c$. What is the maximal ...
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Why open unit ball in any infinite dimensional Banach space is finitely chainable?

In paper "Pointwise products of uniformly continuous functions" by Sam B. Nadler, Jr., He defined the finitely chainable as followings : Let $(X,d)$ be a metric space. An $\varepsilon$-chain in ...
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Approximation of acontinuous function

How to approximate a continuous function on $[-\pi,+\pi]$ which is $2\pi$ periodic by a set of trigonometric polynomials in the sup-norm topology?
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is the set of matrices with trace equal equal to zero compact

Is it true that the set of all matrices with trace equal to zero a connected and compact subset of the 2*2 matrices over R?
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45 views

Metric Space and Open Sets

I'm having trouble figuring out where to go with this problem. Any hints or strategies would be appreciated. I have just the basic definitions for open sets, distance metrics, etc. Consider $\Bbb ...