Tagged Questions

Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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Does there exist non-compact metric space $X$ such that , any continuous function from $X$ to any Hausdorff space is a closed map ?

I know that there is a topological space $X$ which is not compact but such that , for any Hausdorff topological space $Y$ , any continuous function $f:X \to Y$ carries closed sets to closed sets . I ...
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Let $X$ be the union of axes is it homeomorphic to a line, a circle, a parabola or the rectangular hyperbola $xy = 1$?

Let $X$ be the union of axes given by $xy = 0$ in $\Bbb R^2$ . Is it homeomorphic to a line, a circle, a parabola or the rectangular hyperbola $xy = 1$? If we remove the origin from the union of axes ...
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Evenly Spaced Integer Topology is Metrizable

Fustenborg's proof uses an evenly spaced integer topology on $\mathbb Z$ which declares that a basis of open sets as those of the form $a + b \mathbb Z$ (i.e. arithmetic progressions). I'm interested ...
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Is there a metric proof for the Caristi theorem?

I'm writing a paper about hyperconvexity in metric spaces and came across Caristi's theorem: Let $(X, d)$ be a complete metric space and $\phi\colon X \to \mathbb{R}^+$ a continuous function. An ...
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Show that a Cauchy sequence has a fast-Cauchy subsequence

A sequence $\{x_j\}$ is said to be fast-Cauchy if $\sum_1^\infty d(x_j,x_{j+1})<+\infty$. Show that every Cauchy sequence has a fast-Cauchy subsequence. **My attempt:**Argue by contradiction, ...
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How many degrees of freedom are in a flat metric and how does one count them?

I think that there are zero degrees of freedom in a flat metric, but I do not know how to count them. I know that any symmetric metric tensor has $n(n-1)/2$ degrees of freedom, where $n$ is the ...
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How to find an open ball for a metric space?

I don't understand the process to find the open ball. I understand the definition and I understand that for B(0, delta), I need to substitute x as 0. After this stage, I don't understand where to go ...
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Help needed in a proof to show path connected-ness of a special kind of set in real Euclidean space

Let $A$ be a connected subset of $\mathbb R^n$ and $\epsilon >0$ , consider $V:=V(\epsilon , A):=\{x \in \mathbb R^n : d_A(x)< \epsilon \}$ , I need to show that $V$ is path connected , so we ...
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Completion of a sequence space

Let $F$ be a field with some absolute value $|\cdot|$. Consider the space $X$ of sequences $\mathbf{a} = (a_1, a_2, a_3, \cdots)$ for which $a_i \in F$ for all $i\in\mathbb{N}$ and at most finitely ...
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Prove the triangle inequality for d(x,y) = min(|x−y|,1−|x−y|)

Let X be the set [0,1). Define a non-standard metric on X as follows: For two numbers x,y ∈ X, take d(x,y) = min(|x−y|,1−|x−y|). Show that this is a metric. In order to show this is a metric, I need ...
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Which complete weighted graphs are obtained from finite metric spaces?

Let $(X, d)$ be a finite metric space with $X = \{x_1, \dots, x_n\}$. We can associate to this metric space a complete weighted graph with vertices labelled by the points of $X$, and edges weighted by ...
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Order and Metric

Consider the Reals as a totally ordered set via its natural order ( linear continuum ). Such order induces an order topology ( basis is the collection of open-intervals of that order), which ...
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Given a sequence of compact sets $K_{i}$ in $\mathbb{R^{n}}$ and a compact set $K$ in $\mathbb{R^{n}}$, which satisfy the following 2 conditions. $\forall$ $x$ $\in$ $K$, $\exists$ $x_{i}$ $\in$ $K_{... 1answer 191 views Why does countable compactness imply compactness on metric spaces? By "$E$is countably compact", I mean that every countable open cover of$E$has a finite subcover. By "$E$is compact", I mean that every open cover of$E$has a finite subcover. Let$M$be a metric ... 3answers 182 views is union of nested compact spaces still compact? Stel$D$a metric space. Let$K_1 \subset K_2 \subset K_3 \subset ...$a serie of compact sets in$D$. I was wondering if$K = \bigcup_{n=1}^\infty K_n$is compact too. If we take an open cover of$K$... 1answer 68 views Total order and its order topology I noticed that the natural order of the Reals alone, being complete ( satisfying LUB ) , is able to prove that the induced order topology is complete ( every cauchy sequence converges ). We are ... 1answer 106 views Proving set of bounded continuous functions is an open set appreciate your help with the below: Question: Let C[0,1] be the set of continuous functions from [0,1] to$\mathbb{R}$. Consider the metric space M = (C[0,1],d) where d denotes the sup metric. ... 1answer 37 views The distance distribution from the mean for an n-dimensional normal(Gaussian) distribution Let's say we have an n-dimensional normal distribution with identity covariance matrix and 0 mean. When we draw random points in this distribution, how do I get the distribution of the distance from ... 0answers 64 views If$E= A\cup B \cup C$and$E$is connected , where$A$and$B$are disconnected and$C$is connected, then$A \cup C$is connected. If$E= A\cup B \cup C$and$E$is connected in a metric space$(X,d)$, where$A$and$B$are disconnected and$C$is connected, then$A \cup C$is connected. If we consider that$A \cup C$is not ... 1answer 476 views Show that the discrete topology on$X$is induced by the discrete metric Let$X$be a set. Show that the discrete topology on$X$is induced by the metric$d(x, y) = \left\{ \begin{array}{ll} 1 & \mbox{if } x \neq y \\ 0 & \mbox{if } x = y \end{array} \...
Suppose X is a separable metric space and ($U_α$ : α < γ) is an increasing sequence of open sets (i.e. $U_α$ ⊆ $U_β$ for α < β). Show that there is a countable $γ_0$ such that $U_α$ = $U_β$ for ...