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 the length of $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ with the metric $g=\frac{dx^2+dy^2}{y^2}$ and compare with euclidean metric

Consider the metric $g=\frac{dx^2+dy^2}{y^2}$ on $\mathbb{R}_+^2=\{(x,y) \in \mathbb{R}^2 : y>0\}$. Calculate the length of the curve $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ and compare ...
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46 views

All lines of $\mathbb R^3$ are isometric to $\mathbb R$

I have just started reading Metric Spaces by Michael Searcoid. The first Chapter states a result : Suppose $n \in \mathbb N~\forall~ i \in \mathbb N_n,(X_i,\tau_i) $ is a non empty metric space. ...
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48 views

The shortest path in a metric space with a given metric

My questions seem to be very basic and intuitively correct but I can't formally prove them. Before learning metric spaces, for $R^2$, we always define the distance between 2 points as $d_2 = ...
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70 views

Is there a name of such functions?

Let $U$ be an open subset of $ \mathbb R^n$ and consider $f :\mathbb R^n \to \mathbb R$ with the properties that $ f( \partial U)=0$ and $f$ takes negative values on $U$. My questions: Is there ...
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25 views

Boundary and limit points

Suppose that $\Omega \in \mathbb{R}^n$. Prove that if $\vec{x} \notin \Omega$ and $\vec{x}$ is a boundary point of $\Omega$, then $\vec{x} $ is a limit point of $\Omega$. My try: $\vec{x}$ is a ...
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39 views

Prove that if $(f_n)_{n \geq 1} \to f$ uniformly then $f_n \to f \in B(X)$

Let $B(X)$ be the set of bounded functions from $X$ to $\mathbb{C}$ where $X$ is any metric space. Let $(f_n)_{n \geq 1} $ be a sequence in $B(X)$. Show that if $f_n \to f$ uniformly where ...
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172 views

A finite set is closed

Question: Prove that a finite subset in a metric space is closed. My proof-sketch: Let $A$ be finite set. Then $A=\{x_1, x_2,\dots, x_n\}.$ We know that $A$ has no limits points. What's next? ...
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4answers
49 views

Transferring the usual metric from $\mathbb{R}$ to $(0, 1)$ gives us a complete metric space on $(0, 1)$?

I'm watching this video here - https://www.youtube.com/watch?v=zcAvVTFUxS8 The lecturer says that $\mathbb{R}$ and $(0, 1)$ under the usual metric are homeomorphic yet $\mathbb{R}$ is complete and ...
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1answer
17 views

Examples of uniformly discrete proper metric spaces which are not countable.

By uniformly discrete I mean there exists a $C > 0$ such that for all $x \neq y$ we have $d(x, y) \geq C$. By proper I mean the preimage of every closed ball is compact. Are there any examples of ...
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26 views

Proof of triangular inequality for $|z_1-z_2|$

I want to prove that the following is a metric on $\Bbb C$. $|z_1-z_2|$ where $z_1,z_2\in \Bbb C$. I have done all easily but triangular inequality: For triangular inequality, I applied it, as if it ...
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1answer
48 views

Prove that the function in $[0,\pi]$ defined by $f(x)=\sin(x)/x$ and $f(0)=1$ is a contraction

Let $f$ be a function $f:[0,\pi]\to\mathbb{R}$ such that: $$f(x)=\left\{ \begin{array}{ll} \frac{\sin(x)}{x} & \mbox{if } x \neq 0 \\ 1 & \mbox{if } x = 0 \end{array} \right.$$ I want ...
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1answer
23 views

For each $n \in \mathbb{N}$ consider $A_n = \{1/n\}\times [0,1]$ and let $X=\bigcup_{n \in \mathbb{N}}A_n \cup \{(0,0)(0,1)\}$

For each $n \in \mathbb{N}$ consider $A_n = \{1/n\}\times [0,1]$ and let $X=\bigcup_{n \in \mathbb{N}}A_n \cup \{(0,0)(0,1)\}$. Show that $\{(0,0)\}$ and $\{(0,1)\}$ are connected components. ...
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1answer
22 views

Is this a metric on the shift space?

Consider $$ \Omega_N:=\left\{\omega=(\ldots,\omega_{-1},\omega_0,\omega_1,\ldots): \omega_i\in\left\{0,1,\ldots,N-1\right\}\text{ for }i\in\mathbb{Z}\right\} $$ and $$ ...
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240 views

Let $D$ be a bounded domain (open connected) in $ \mathbb C$ and assume that complement of $D$ is connected.Then show that $\partial D$ is connected

I am trying to prove the following famous result in Point Set Topology. Let $D$ be a bounded domain (open connected) in $ \mathbb C$ and assume that complement of $D$ is connected. Then show that ...
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1answer
28 views

What does $\overline B(0,1)\subseteq (\mathscr{C}([0,1],\mathbb{C}),d_\infty)$ mean?

What does $\overline B(0,1)\subseteq (\mathscr{C}([0,1],\mathbb{C}),d_\infty)$ mean? Could someone explain the meaning of this? How can a closed ball be a subset of a metric space of continous ...
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1answer
64 views

Generalization of metric spaces?

Usually we define a metric on a space $X$ to be a map $X\times X\to\mathbb{R}$ that satisfies a few axioms. $\mathbb{R}$ has of course a total order. What if we instead have a metric $X\times X\to A$ ...
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1answer
67 views

Is the completion of a metrizable topological group metrizable?

Let $G$ be a topological group and its two-side uniformity $\mathcal{U}$ (that is the uniformity generated by right uniformity and left uniformity of $G$) coincides with the uniformity of a metric ...
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1answer
37 views

Show that a metric space has a countable base if and only if each open cover has a countable subcover

Show that a metric space has a countable base if and only if each open cover has a countable subcover. Proof $\Rightarrow$ using Lindöf's lemma But how does one show $\Leftarrow$? Consider $\{ ...
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2answers
23 views

Locally connected metric spaces

I have the following definition of a locally connected metric space. Given $(X,d)$ a metric space, $x \in X$ and given $U \ni x$ a neighbourhood. Then there exist a connected neighbourhood $V$ susch ...
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1answer
50 views

Error in the reasoning?

Give an example of a metric space $(M,d)$, an $a\in M$ and a $R\in \mathbb{R}, R> 0$ such that $$\overline{B(a,R)} \not = \overline{B} (a,R)$$ Prove in general ...
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1answer
36 views

Every metric space contains a discrete, coarsely dense subset

I'm wondering on how to prove the following: Let $(X,d) $ be a metric space. We say that a subset $ A \subset X $ is coarsely dense iff $ \exists_{C > 0} \forall_{x \in X} \exists_{a \in A} d(x,a) ...
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65 views

Closest packing of equal balls in $\Bbb{R}^4$

I know how to find the closest packing of equal spheres in $\Bbb{R}^3$. I'd like to know how to find the closest packing of equal balls in $\Bbb{R}^4$ with the standard Euclidian metric. I suspect ...
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0answers
22 views

Prove that $\frac{d(a,b)}{1 + d(a,b)}$ is a metric? [duplicate]

Given any metric $d(a,b)$ where $a,b\in\mathbb{R}$, prove that $$d_1(a,b):=\frac{d(a,b)}{1+d(a,b)}$$ is also a metric. We have $0 \leq s\leq t \Rightarrow \frac{s}{1+s} \leq \frac{t}{1+t}$ to support ...
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2answers
22 views

Open set in a metric space is union of closed balls?

We know that every open set $A$ is in a metric space $(X,d)$ is the countable union of closed sets, and every open set $A$ is in a Euclid space $R^n$ is the countable union of closed balls. My ...
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1answer
26 views

Inequality involving distance between two points

Let $\Omega$ be a set in $\mathbb{R}^n$ Fix $x\in\Omega$, and $p\notin \Omega$ Then, is it always true that $$\|x-p'\| \leq \|x-p\|$$ , where $\|\cdot\|$ is the Euclidean norm, and $p'$ is the ...
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45 views

Show that $\mathbb{R}$ and $\mathbb{R}^n$ are not homeomorphic if $n\geq 2$

Show that $\mathbb{R}$ and $\mathbb{R}^n$ are not homeomorphic if $n\geq 2$. I want to use a connection-type argument. I thought of giving the following proof; Suppose that there exist such a ...
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1answer
29 views

The space of absolutely convergent series is complete

For clarity: $$ l^{1}(\mathbb{N}) = \left\{ (x_{n})_{n} \ \middle|\ \sum_{n}|x_{n}| \in\mathbb{R} \right\} $$ $$ d_{1}:\ l^{1}(\mathbb{N}) \times l^{1}(\mathbb{N}) \rightarrow \mathbb{R}^{+}:\ ...
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1answer
19 views

How to convert table into a distance function?

Been stumped on this past paper question for a while, it's in the context of clustering and the next part is using single linkage bottom-up hierarchical clustering to form a dendrogram using your ...
0
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1answer
37 views

Formal name for the coordinate values of the pushforward of the inverse metric on an embedded manifold?

What is the formal name of the following object: \begin{align}\tag{4} \Delta^{\alpha \beta} = \dfrac{\partial y^\alpha}{\partial x^m} g^{mn} \dfrac{\partial y^\beta}{\partial x^n} \end{align} where ...
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1answer
13 views

A convergent sequence of non-expansions converges uniformly on a totally bounded domain

Here's a theorem that I tried to prove: Let $V,d_V$ and $W,d_W$ be metric spaces and $(f_n)_n$ a sequence of non-expansions that converges to a function $f:A \subseteq V \rightarrow W$: $$ f_n:\ A ...
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1answer
35 views

Can I pull a limit through a metric?

Let $X,d_X$ and $Y,d_Y$ be metric spaces and $(f_n)_n$ a sequence of (continuous) functions. Does this hold and, more importantly, why? $$ d_Y\left(\lim_{n\rightarrow +\infty}f_n(x), ...
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1answer
25 views

Book recomendation for function sequences.

I wanted to study about sequences of functions defined in metric spaces. What book/books do you recommend? Thanks!
3
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58 views

In a metric space $(X, d)$, if closed sets $A$, $B$ contain sequences $a_n,b_n$ such that $d(a_n,b_n)\to 0$, must $A\cap B\neq \emptyset$?

I wrote a test yesterday in which one of the questions asked us to prove that if $A$ and $B$ are disjoint closed subsets of a metric space $(X, d)$, then there exist disjoint open subsets $U$ and $V$ ...
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If $C \subset X$ and $\mathcal{U} \subset X$ is open. Is $C \cap \mathcal{U}$ open in $C$?

In an exercise regarding connection, I came to the following problem, I am given $C \subset X$ and $\mathcal{U} \subset X$ is open (where $(X,d)$ is a metric space). And I could use that is $C \cap ...
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1answer
15 views

Conceptual question regarding conection in metric spaces.

I have to give an example of two sets $A,B \subset \mathbb{R}$ such that both are connected, but $A\cup B$ is not. So I thought of a trivial example $(0,1) \subset \mathbb{R}$, $(2,3) \subset ...
1
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1answer
14 views

Prove that there exist $r>0$ such that $\bigcup_{x \in K} B(x,r) \subset V$

Let $M$ be a metric space, let $K \subset V \subset M$, $K$ compact, $V$ open. Prove that there exist $r>0$ such that $\bigcup_{x \in K} B(x,r) \subset V$ I came up with a proof, but there is ...
0
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1answer
15 views

Distance attained by a function

Let $A$ be a subset of $\mathbb R^n$ and let $x\in \mathbb R^n$. Then $\exists y_0\in A$ such that $d(x,y_0)=d(x,A)$ if $A$ is a non-empty subset of $\mathbb R^n$. $A$ is a non-empty closed subset ...
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35 views

Is this function a metric?

Let $X$,$d$ be a metric space. Define $d'$ as the minimum of $1$ and $d$: $$ d':\ X^2 \rightarrow \mathbb{R}:\ d'(x,y) = \min\{1,d(x,y)\} $$ The question is whether $d'$ is a metric. I've managed to ...
3
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3answers
113 views

$[0,1)$ as a subspace of the Euclidean metric space?

Consider the Euclidian metric space $(\Bbb R,d)$ where $d(x,y)=|x-y|$ is the usual metcic on $\Bbb R$. The set $[0,1)$ is not closed in $(\Bbb R,d)$ but considered as a subspace it is closed by ...
3
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1answer
38 views

Two problems related to continuity of a metric from Munkres' topology book

Let $X$ be a metric space with metric $d$. Show that $d:X\times X\to \mathbb{R}$ is continuous. Let $X^\prime$ denote a space with the same underlying set as $X$. Show that if $d:X^\prime\times ...
3
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1answer
26 views

Compactness of a group with a bounded left-invariant metric

Let $G$ be a group equipped with a left-invariant metric $d$: that is, $(G,d)$ is a metric space and $d(xy,xz) = d(y,z)$ for all $x,y,z \in G$. Suppose further that $(G,d)$ is connected, locally ...
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63 views

Short proof that $\rho^\prime(x,y) = \min\{1,\rho(x,y)\}$ is a metric [duplicate]

Let $(X,\rho)$ be a metric space. Define $\rho^\prime: X \times X \to \mathbf{R}$ by $\rho^\prime (x,y) = \min\{1,\rho(x,y)\}$ for all $x, y \in X$. Does anyone know of a short proof that ...
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2answers
40 views

Compactness of $A\subset \mathbb R$ w.r.t. two different topologies

Let $d$ be the Euclidean metric and $d'$ be any other metric on $\mathbb R$. Let $A\subset \mathbb R$ be a closed and bounded subset with respect to $d'$. Then which is TRUE ? (A) $A$ is ...
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1answer
37 views

A fixed-point theorem by Zamfirescu

I am having a trouble with understanding the proof of a fixed-point theorem by Zamfirescu. Could somebody please explain how the inequality in the inner, pink rectangle is obtained from the previous ...
0
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1answer
58 views

Show that A is an open subset of M

If $m\in\mathbb{N}$, $M=\{0,1\}^{\mathbb{N}}$ and $A \subseteq M$ is an open set of sequence where the number 1 appears at least $m$ times. Show that $A$ is an open subset of $M$. I wanted to show ...
1
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1answer
23 views

Is the closure of a meager set meager?

How to show that the closure of a meager set is meager? I tried like this: Suppose that it is not meager then $cl(A)$, where $A$ is a meager set in a metric space $(X,d)$ contains an interior point ...
0
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0answers
27 views

Closed subsets of empty interior are of 1st category

In a metric space is it true that closed sets with empty interior are of 1st category? I.e., that it can be represented as a at most countable union of meager sets? Thanks
0
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1answer
35 views

Is point-to-set distance function $C^\infty$ for $\mathbb{R}^n$

Let $x\in \mathbb{R}^n$ and $Q\subset \mathbb{R}^n$. Then we define the point-to-set distance function as: $$ d_Q(x) = \inf_{y \in Q} \| x-y\| $$ It's continuous for every normal space (not only ...
0
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0answers
26 views

Hölder's inequality/Cauchy-Schwarz for Bregman Divergence?

Consider the Bregman divergence. $$ D_F(p, q) = F(p)-F(q)-\langle \nabla F(q), p-q\rangle. $$ And its dual norm: $D_{F*}(p, q) $ where $ F^*(y) = \arg\min_x \left\{ \langle x, y\rangle - F(x) ...
3
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2answers
76 views

Motivation for separation axioms

I have recently been studying different separation and countability axioms in topology. I am looking for a motivation for why such a refined division of different axioms was made and is studied. I am ...