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.

learn more… | top users | synonyms (1)

3
votes
1answer
133 views

Equality on functions in $ \mathbb{R}^n $

Let $ f,g : M \subset \mathbb{R}^p \to \mathbb{R}^q $ continuous. Given $ a \in M $, supose that all open ball centered in $a$ contains a point $x$ such as $f(x) = g(x) $. Show that $ f(a) = g(a) $. ...
1
vote
1answer
50 views

Separability of the Space of all Real-Valued functions over $[a,b]$ with a Continuous First Derivative

I'm reading Neal Carothers' Real Analysis and I'm stuck on the following question: Let $f$ be real-valued, continuously differentiable function over $[a,b]$ and let $\epsilon>0$. Show that there is ...
0
votes
0answers
42 views

Both $F$ and $C$ are closed sets but their sum $F+C$ is not closed. [duplicate]

In context to the question what will be an counter example such that both $F$ and $C$ are closed sets in $ \Bbb R^n$ but their sum $F+C$ is not closed in $ \Bbb R^n$?
3
votes
1answer
63 views

Spaces vs. Structures

Examples of spaces I've come across include vector spaces, inner-product spaces, and metric spaces. Examples of structures I've met include rings, fields, and groups. I have always understood spaces ...
2
votes
0answers
35 views

Exponents for Hölder functions on metric spaces

Sometimes people talk about Hölder functions on metric spaces without mentioning the allowed range for the exponent. On manifolds, it's traditional to assume an exponent in $[0, 1]$, since all ...
0
votes
1answer
42 views

Metric spaces - Limit points and Isolated points

I apologise for this, but I have numerous potential misunderstandings. $\Bbb R^2$ is a metric space, since it is a subspace of the Euclidean space, $\Bbb R^n$ I can look at a set $E$ with elements ...
0
votes
3answers
61 views

E is closed if every limit point of E is a point of E?

E is closed if every limit point of E is a point of E? Should that be "E is closed if every point of E, is a limit point"? I don't understand. Limit points are essentially points that hug other ...
-2
votes
1answer
78 views

Understand the definition of convex metric spaces

I am trying to understand the following definition: We call a set $E\subset \Bbb R^k$ convex if>$$\lambda x+(1-\lambda)y\in E$$ Whenever $x\in E, y\in E$ and $0\lt \lambda \lt 1$ Clearly ...
0
votes
2answers
32 views

Understanding part of a theorem of Calculus of Variations

I have trouble understanding the following statement (From Gelfland's Calculus of Variations book): If $\phi[h]$ is a linear functional and if ...
2
votes
0answers
60 views

Density character of a metric subspaces

Is it true that if $M$ is a metric space and $N$ is a metric subspace of $M$ (I mean, $N\subseteq M$ and the metric defined on $N$ is the same metric on $M$ restricted to $N$) then the density ...
2
votes
2answers
45 views

Every $p$-norm ($p \in [0,\infty]$) generates the same class of open sets on $\mathbb{R}^n$

The following claim has been made in my multivariable analysis class, and I think I have the idea of the proof but I can't quite seem to get down to the rigorous proof the instructor wants: Every ...
0
votes
1answer
61 views

Why isn't the completion of $C^0$ wrt. the $L^2$ norm a space of sequences instead of a space of functions?

We know that $L^2(\Omega)$ can be defined as the completion of $C^0(\Omega)$ with respect to the norm $$\left(\int_\Omega |u|^2\right)^{\frac 12}.$$ But strictly speaking, $L^2(\Omega)$ is a space of ...
0
votes
3answers
58 views

To show that $X = (0,1]$ is complete .

Show that $X = (0,1]$ is complete with respect to the metric $e $ where $e(x,y) = |\frac{1}{x} - \frac{1}{y}|$. My proof: let $(x_n)$ be Cauchy in $(X,e)$. Let $(t_n) := \frac{1}{(x_n)}$. Then ...
1
vote
1answer
105 views

Does topological equivalence of metrics imply strong equivalence?

I know that if $(X,d_1)$ and $(X,d_2)$ are metric spaces and for some positive constants $a,b$ , $ad_1(x,y) \le d_2(x,y) \le b d_2(x,y) $ for every $x,y$ in $X$ , then a subset $A$ of $X$ is $d_1$ ...
1
vote
1answer
46 views

How to prove this statement?

I cannot prove this proposition directly . Let $(X,d)$ and $(Y,d')$ be metrice spaces. Let $f$ be a function from $X$ to $Y$. If $\overline{f^{-1} ( B)} \subseteq f^{-1}( \overline B)$ for all ...
2
votes
3answers
35 views

To show that $d $ and $ e$ are equivalent.

On the set $X = (0,1]$, consider the usual metric $d(x,y) = |x-y|, (x,y \in X) $ and another function $e: X\times X \to R$ given by $e(x,y) = |\frac{1}{x} - \frac{1}{y}|$. Show that $d $ and $ e$ ...
2
votes
3answers
132 views

Prove the intersection of a compact set and a set with no accumulation points is finite

Let $S\subset\mathbb{C}$. We say that $z_0$ is an accumulation point of $S$ if for every $r>0$, the intersection $D(z_0,r)\cap S$ is an infinite set. Let $U\subset\mathbb{C}$ be an open set such ...
1
vote
1answer
58 views

Open set in Hilbert Cube.

Any open set in the Hilbert Cube is the union of open subsets of the form $$U_1 \times ... \times U_n \times X_{n+1} \times .... \times X_{n+k} \times...$$ where $X_k := [0, \frac{1}{k}]$ for $k \in ...
6
votes
1answer
35 views

Let $A$ be any subset of $\mathbb R^{+}$ , then there exist a metric space $(X,d)$ such that $d:X \times X \to A \cup \{0\}$ is a surjection?

Let $A$ be any subset of the set of positive real numbers $\mathbb{R}_+$ ; then does there exist a metric space $(X,d)$ such that $d\colon X \times X \to A\cup\{0\}$ is a surjection ?
0
votes
0answers
35 views

Equivalence of Definitions of completion of metric space

I've come across two different definitions for a completion of a metric space and am trying to figure out why they are equivalent. The definitions are: 1) Let $(X,d)$ be a metric space. Then ...
1
vote
0answers
33 views

Rationals in an interval $[a,b] \in \Bbb R$

(i) For which real values $a$ and $b$, ($a < b$), is the set $[a,b] \cap \Bbb Q$ open in $(\Bbb Q, d)$, (where $d(x,y)= \lvert x-y \rvert$)? (ii)For which real values $a,b$ is the set $[a,b] \cap ...
1
vote
2answers
54 views

for which values of $x,y$ is $[x,y]\cap \mathbb{Q}$ closed?

for which values of $x,y$ is $[x,y]\cap \mathbb{Q}$ closed in the metric space $(\mathbb{Q},d)$ where $d(x,y) = |x-y|$ my attempt: I suspected it's closed for all real numbers: let $x,y \in ...
0
votes
0answers
280 views

The triangle inequality for shortest paths of graphs

In why-the-triangle-inequality I found the statement: for example if $d(a,b)$ measures the "length" of the "shortest path" between points $a$ and $b$ (and this can be interpreted quite ...
1
vote
1answer
46 views

Question in analysis: subset of open interval in $\Bbb R$

Consider metric space $(X,d)$, $X=(a,b)\subset \Bbb R$, $d(x,y)= \lvert x-y \rvert$. Let a subset $S \subset (a,b)$ be open and closed. Show that either $S=(a,b)$ or $S= \emptyset$. There's a ...
1
vote
1answer
88 views

Determine if a function is a metric

I have been asked the following question in one of my tests. I'm not sure of how to do it. Consider the plane $X = \Bbb R^2$. For each of the following two proposed distance functions, determine ...
0
votes
1answer
34 views

Non-Lipschitz homeomorphism from compact metric space to itself

Is it possible to find a compact metric space $(X,d)$ with more than one point and a homeomorphism $\varphi:(X,\tau) \to (X,\tau)$ where $\tau$ is the topology induced by $d$ such that $$(\forall N\in ...
0
votes
2answers
41 views

The infimum $\inf_{(a,b) \in A\times B} \; \rho(a,b)$ is attained for any two compact sets $A,B$

Let $A,B$ be compact sets in $(S,\rho)$. Define $\rho(A,B)$ by $$\rho(A,B) = \inf_{(a,b) \in A\times B} \; \rho(a,b)$$ Show that there exists $a_0 \in A, b_0 \in B$ s.t. $$\rho(A,B) = \rho(a_0,b_0)$$ ...
2
votes
2answers
90 views

Direct sum of metrizable spaces.

I managed to prove that an arbitrary direct sum of metrizable spaces is again metrizable. However, I used the theorem that says that a hausdorff regular space is metrizable if and only if there existd ...
-1
votes
2answers
35 views

prove of topology and metric spaces [closed]

Prove or disprove $f: A \to B$ a function from $A$ to $B$. $A_i$ subset of $A$ and $B_i$ subset of $B$. If $A_0 \subset A_1$ then $f(A_0) \subset f(A_1)$ $f(A_0 \cup A_1) = f(A_0) \cup f(A_1)$ ...
0
votes
1answer
20 views

showing restricted metric still forms a complete metric space

Let $(A,d)$ be complete. Let $B$ be a closed subset of $A$. Then show the metric space $(B,d|_{B\times B})$ is complete. I have shown from a theorem that $(B,d)$ is complete, but I am not sure how to ...
1
vote
2answers
69 views

Let $a$ and $b$ be arbitrary real numbers with $a < b$. Show that $[a,b]$ is closed by proving its complement is open.

Let $a$ and $b$ be arbitrary real numbers with $a < b$. Show that $[a,b]$ is closed by proving its complement is open. I don't have any idea on this, can anyone help me on this?
1
vote
1answer
59 views

Prove that, for $x \in \mathbb R$ and $\delta_x > 0$, the open interval $(x-\delta_x, x+\delta_x)$ is itself an open set [duplicate]

Prove that, for $x \in \mathbb R$ and $\delta_x > 0$, the open interval $(x-\delta_x, x+\delta_x)$ is itself an open set. I am preparing for my exam and we will be asked to prove various ...
1
vote
0answers
79 views

Entropy of isometric extension

A similar question to mine was asked before at the address below but it was not answered there so I am asking it again. Also there is a more specific case I am interested in. Topological entropy of ...
0
votes
2answers
63 views

Show that $d_V$ is a metric

Problem: For points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ in $\mathbb{R}^2$ define: $d_V(p,q) = \begin{cases}1 & p_1\neq q_1 \ or\ |p_2 - q_2|\geq 1 \\ |p_2 - q_2| & p_1= q_1 \ and\ |p_2 ...
1
vote
1answer
53 views

How to show that this set is closed in $\mathbb{R}^n$?

For an open set $\Omega\subseteq\mathbb{R}^n$, let $K_j$ be the set of points $x$ of $\Omega$ such that $\text{dist}(x,\partial\Omega)\geq1/j$ and $|x|\leq j$. Question : Why is $K_j$ closed ? ...
3
votes
2answers
162 views

Construct a set of real numbers whose limit points comprise the set of integers $\mathbb{Z}$

My thought process is the following: Let $S=\{ m + \frac{1}{n}| m \in \mathbb{Z},n \in N \}$. Then I need to show that the limit points of $S$ are indeed the integers and that these are the only ...
0
votes
1answer
48 views

Statiscal Distance Properties

Please anyone could give me any idea of how prove the following property of statistical distance: $d(AB,CD)\leq d(A,C)+d(B,D)$ Remenber that: $(X,d)$---> Metric Space $d:X\times X\rightarrow ...
0
votes
4answers
107 views

Proving $d_1$ is a metric.

If $d$ is a metric on a set X, then $d_1 = \frac{d(x,y)}{1+d(x,y)}$ is also a metric. I have proved the other conditions of being a metric except the triangle inequality. Please help!!
1
vote
1answer
74 views

Sequential continuity on metric spaces

Please give me a hint for proving this statement: Let $(X,d)$ and $(Y,d')$ be metric spaces, $f$ a function from $X$ to $Y$. If $f^{-1}(B) $ is closed in $X$ for all closed subset $B$ of $Y$, then ...
0
votes
1answer
46 views

Computing $\sup_{\left \| u \right \|=1} d(u,F)$ in a closed subspace of a normed space.

I have come across with the following problem: Let $E$ be a normed space and $F$ a closed subspace of $E$. It's asking to compute $\sup\limits_{\left \| u \right \|=1} d(u,F)$. What I know it's ...
-1
votes
3answers
81 views

Exercise on Metric space

I hve this exercise it is very simple but i don't know how to write the answer Let $A$ be a nonempty set in $(E,d)$, for $\varepsilon>0$ we note $$V_{\varepsilon}(A)=\{x\in E, ...
0
votes
1answer
239 views

Separability of the Set of Bounded Functions over [0,1]

I'm working through Neal Carothers' Real Analysis and I'm stuck on trying to show that the set $B$ of bounded, real-valued functions over $[0,1]$ is not separable. The metric of this set is ...
4
votes
2answers
90 views

In a metric space we have $B(x,r) = B(y, s)$, is it necessary that $x = y$ and $r = s$??

If in a metric space we have $B(x,r) = B(y, s)$, is it necessary that $x = y$ and $r = s$? I think that the center of the balls i.e. $x$ and $y$ must be same but the radius $r$ and $s$ may not be ...
0
votes
1answer
50 views

Is there a Heine cretierion of liminf of a function?

Lately i've been struggling with understanding the meaning of $\liminf_{x\to x_0}f(x)$ assuming $f:X\to\mathbb C$ for $X$ a metric space, or for that matter $f:\mathbb R\to \mathbb R$. Could you give ...
1
vote
0answers
26 views

Connectedness ( cardinality and connectedness) [duplicate]

$(X,d)$ metric space and $A\subset X$ and $A$ is connected. $$ \text{Card}(A) > 2 \implies \text{Card}(A) \geq \text{Card}(\mathbb{R}).$$ How do I prove it ?Waiting for your help?
1
vote
2answers
50 views

Prove this function is lower semi-continuous

Let $X$ be a metric space, and $B$ his borel $\sigma$-algebra. Fix $r>0$ Let $\mu$ be a probability measure on $(X,B)$ and define $f(x)=\mu(B(x,r))$. Show that $f$ is lower semi continuous. What ...
0
votes
1answer
34 views

What is an example of a connected subset of $\mathbb{R}^2$ where the interior is not connected?

In $\mathbb{R}^2$ with the usual metric, could this be an open disk, e.g. $ \{(x,y) : x^2 + y^2 \le 2\}$? Thanks in advance!
2
votes
1answer
102 views

Net convergence in metric spaces

This is a question about convergence of nets which I don't quite understand yet. In metric spaces convergence of sequences encodes the topology but suppose we want to study convergence of nets even ...
2
votes
1answer
90 views

Showing that every set in a metric space is open and closed

Define: $$d(x,y) = \frac{|x-y|}{1+|x-y|}$$ I have shown that $(\mathbb{N}, d)$ Is a metric space. I have also been asked to describe all open balls of radii $1/3, 2/3 $ and $3$ in this metric space. ...
1
vote
1answer
68 views

Is it possible for an uncountable subset $S$ of $\Bbb R$ to satisfy $∂S = S$?

Is it possible for an uncountable subset $S$ of $\Bbb R$ to satisfy $∂S = S$? Please shed some light to it. I am not getting any clue to it.