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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Continuty of functions inside a open ball

Let $ f: X \subset \mathbb{R}^p \to \mathbb{R}^q $ and $ a \in X$. Supose that for all $ \epsilon > 0 $ exists $ g: X \to \mathbb{R}^q $ continuous at $a$ such as $ \| f(x) - g(x) \| < \epsilon ...
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Computation of the extrinsic curvature tensor for a warp drive metric.

In Miguel Alcubierre's renowned paper discussing a "warp drive" metric, he discusses the extrinsic curvature. Here is an extract. My questions are quite trivial to someone who understands the ...
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21 views

Closure, interior and boundary of $(0, 1)$ with Zariski topology

If we consider $\mathbb{R}$ together with the Zariski topology, what is the closure, interior and boundary of $(0,1)$? A set is closed iff it is either finite or $\mathbb{R}$ under this topology, so ...
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3answers
21 views

Compact Sets Metric Spaces

Lets $(\Bbb R,|x-y|)$ be a metric space. By the Heine-Borel theorem, it obviously follows that $\Bbb Q$ is not a compact set. Now, if I were to consider $\Bbb Q \cap[-1,0]\subset\Bbb R$ is that a ...
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1answer
18 views

Proof that the composition of two contractions on the same metric space (X,d) is also a contraction

I am required to prove that given the metric space $(X,d)$, a contraction $T : X \to X$ and another contraction $S : X \to X$, the compositions $T \circ S$ and $S \circ T$ are also contractions. I ...
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1answer
8 views

To show $\{(x,y,z) : x+z^2\sin(x+y) \ge z \}$ is closed in $\mathbb R^3$ by elementary methods

How to show that the set $\{(x,y,z) : x+z^2\sin(x+y) \ge z \}$ , where $x,y,z$ each are from the set of real numbers , is a closed subset of $\mathbb R^3$ under usual Euclidean metric ? I know how to ...
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1answer
21 views

Function continuity outside a closed subset

Let $f:M \subset \mathbb{R}^p \to \mathbb{R}^q $,continuous at $a \in M $. Show that if $f(a) \notin \overline{B} (b,r) \subset \mathbb{R}^q $, then exists $ \delta > 0 $ such as $ f(x) \notin ...
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1answer
21 views

Show that $d_V(x,y)$ is metric

Question: On the set of integers $\mathbb{Z}$, show that the function $d$, defined as follows, is a metric: $$d_V(x,y) = \begin{cases} 0 & \text{if}\ x=y \\ \min{\{\dfrac{1}{n!}}\}\mid\ n!\ ...
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1answer
35 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) $. ...
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1answer
10 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 ...
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36 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$?
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1answer
42 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 ...
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14 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 ...
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1answer
15 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 ...
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1answer
49 views

If $F$ is a closed subset of $\Bbb R^n$ and $x \in \Bbb R^n$ then $x+F$ is closed.

If $F$ is a closed subset of $\Bbb R^n$ and $x \in \Bbb R^n$, is $x+F$ still closed? Can you generalize this question? I showed that $x+F$ is closed but what will be the generalization of the ...
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3answers
26 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 ...
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1answer
59 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 ...
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2answers
28 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 ...
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0answers
26 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 ...
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2answers
33 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 ...
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1answer
22 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 ...
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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 ...
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1answer
11 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 $d_1$ ...
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1answer
38 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 ...
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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$ ...
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3answers
42 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 ...
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When is the completion of a topological vector space a Frechet space?

Suppose $X$ is a topological vector space with the metric topology. If we take the completion of $X$ with respect to the metric, will we get a Frechet space? Are there any extra conditions needed to ...
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1answer
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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 ...
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1answer
29 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 ?
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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 ...
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21 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 ...
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2answers
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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 ...
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39 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 ...
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1answer
37 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 ...
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1answer
29 views

Prove that $(0,1)\times (0,1)$ is open in $\mathbb R^2$. [closed]

Consider a plane $\mathbb R^2$ with the metric $$d(x, y) = \sqrt{|x_1 - x_2|^2 + |y_1 - y_2|^2}.$$ Show that $U = (0,1) \times (0,1)$ is an open set in $\mathbb R^2$ under this metric. How to ...
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1answer
42 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 ...
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1answer
17 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 ...
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2answers
33 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. $$\rho(A,B) = \rho(a_0,b_0)$$ ...
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2answers
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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 ...
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2answers
31 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 $f(A_0) \subset f(A_1)$ $f(A_0 \cup A_1) = f(A_0) \cup f(A_1)$ ...
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1answer
39 views

Question about a topology proof [closed]

Hi. I need help with this simple question. I am not able to get this one.
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0answers
21 views

Let $Q \subset (m,n)$ be a subset which open and closed, show that $ Q = (m,n)$ or $ Q = \emptyset$

Consider the metric space $((m,n),d)$ where $(m,n) \subset \mathbb{R}$ and $d(x,y) = |x-y|$ Let $Q \subset (m,n)$ be a subset which open and closed, show that$ Q = (m,n)$ or $ Q = \emptyset$ there ...
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1answer
12 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 ...
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2answers
42 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?
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0answers
49 views

Help with general topology questions [closed]

Given $P_0=(x_0,y_0)$ and $P_1=(x_1,y_1)$ points in $\mathbb{R}^2$, define the distance between $P_0$ and $P_1$ as $$d(P_0,P_1)=\sqrt{(x_0-x_1)^2+(y_0-y_1)^2}.$$ In $\mathbb{R}^2$, the equivalent of ...
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3answers
56 views

Proof of questions with general topology. [closed]

Let $A$ be any subset of $\Bbb R$ with $|A| < \infty$. Prove that $A$ is closed. Can anyone please help me with this proof?
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votes
3answers
42 views

doubt with proof in genral topology [closed]

let Z and Q represent the integers and the rationals, respectively. prove that Z is a closed subset of R. Frankly I don't have an idea how to start. Can anyone please help me with this proof.
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vote
1answer
48 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 ...
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0answers
19 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 ...
2
votes
2answers
51 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 ...