Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

1,805 questions
10k views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
3k views

Is there a categorical definition of submetry?

(Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ...
316 views

395 views

Can you build metric space theory without the real numbers?

Every metric space $M$ has a completion $\widehat M$, that is, it can be embedded as a dense subset in a complete metric space. When I first came across this theorem, I thought "well, that's amazing, ...
203 views

110 views

Is the Hausdorff distance between $I=[0,1]$ and $B_n = \{0, \frac{1}{n}, \cdots, \frac{n-1}{n}, 1\}$ $0$ or $\frac{1}{2n}$?

In this post, it is claimed that the Hausdorff distance $d_H$ between $I=[0,1]$ and $B_n = \{0, \frac{1}{n}, \cdots, \frac{n-1}{n}, 1\}$ equals zero. Is it not equal to $\frac{1}{2n}$ ? For ...
266 views

Find distance to a set (subspace) without computing closest point

General setup: we have a finite-dimensional normed linear space $(V, \| \cdot \|)$, a subspace $U \subset V$, and a fixed vector $v_0 \in V$. We want to find the distance between $v_0$ and $U$. (No ...
210 views

A discontinuous function $f: X \rightarrow Y$ satisfying: for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$

Find a function $f: X \rightarrow Y$ between metric spaces $X$ and $Y$ that is not continuous but has the property that for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$ Solution Attempt: ...
462 views

What do the eigenvalues/vectors of a metric describe?

Given a finite metric space $(X = \{ x_i \}_{i=1}^n,d)$, one can form the matrix $A$ of pairwise distances $a_{ij} = d(x_i, x_j)$. What does the eigenspectrum of this matrix say about the metric $d$? ...
541 views

94 views

Let us first define some terms. Definition of Pre-pseudometric Let $X\ne\emptyset$ and a function $\varphi:X\times X\to\mathbb{R}$ will be called a pre-pseudometric on $X$ if, $x=y\... 0answers 101 views An open ball in$C[0, + \infty)$Consider the space$C[0, +\infty)$of all continuous, real-valued functions on$[0, + \infty)$with metric $$d( \omega_1, \omega_2 ) = \sum_{n=1}^{\infty} \frac{1}{2^n} \max_{t \in [0,n]} ( \min \... 0answers 112 views Defining a metric in the tangent spaces T_xM I'm working with a metric D over the manifold of grassman G_n(\mathbb{R}^{d}) and have difficulties to extend D. Let me explain: If M is a submanifold of dimension n in \mathbb{R}^{d} ... 0answers 105 views Which complete weighted graphs are obtained from finite metric spaces? Let (X, d) be a finite metric space with X = \{x_1, \dots, x_n\}. We can associate to this metric space a complete weighted graph with vertices labelled by the points of X, and edges weighted by ... 0answers 179 views Order and Metric Consider the Reals as a totally ordered set via its natural order ( linear continuum ). Such order induces an order topology ( basis is the collection of open-intervals of that order), which ... 0answers 409 views Sufficient conditions for closed infinite pasting lemma It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for X = \bigcup X_i with X_i closed in X such that \left. f\... 0answers 713 views Completeness of a metric space with the Hausdorff metric Let (Y,d) be a metric space and let K(Y) denote the set of all non-empty compact subsets of Y. This collection is a metric space when equipped with the Hausdorff distance h. I want to prove ... 0answers 5k views Proving the l_p space is complete. I'm trying to prove l_p spaces are complete. We have an l_p space W. Let us take a cauchy sequence. There exists N_0\in\Bbb{N} such that for m,n>N_0, d(x^m,x^n)<\epsilon. This ... 0answers 48 views Is the metric space of Lipschitz Function with d_{\infty} complete? This is my real analysis homework. Define$$\mathcal{L} = \{f : [a,b] \to \mathbb{R} : f \text{ is a Lipschitz function }\}$$and the metric$$d_{\infty}(f,g) = \sup\{|f(x)-g(x)| : x \in [a,b]\}$$... 0answers 44 views Uncountable collection of open sets. Can there be an uncountable collection of open sets in$\Bbb R^n$? My idea: Since every open set contains at least one rational number, I can match each open sets to rational numbers and rational ... 0answers 37 views Valid metric on a hyperbolic space Note: cross-posted to mathoverflow.net I'm looking at the distance that's defined in this paper on Poincaré Embeddings:$d(\mathbf{u}, \mathbf{v}) = \operatorname{arccosh} \left(1 + 2\frac{\left\| \...
Let $M$ be some metric space. We will also say that $M$ is a measurable space. How do you find a probability measure $P$ on $M$ that maximizes the expected distance between two points? (That is, $P$ ...