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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Doubt in Spivak's examples of Manifolds
I've started to study Differential Geometry in Spivak's first volume of his Differential Geometry books. I like very much his approach since general topology isn't assumed, and since he gives many ...
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58 views
A bijective mapping between metric spaces is open iff it is closed
Let $X$ and $D$ be metric spaces and suppose that $f: X \to D$ is one-one and onto. Show that $f$ is an open map iff $f$ is a closed map.
how can I able to solve this problem
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1answer
110 views
Unit ball of a Separable Banach Spaces is metrizable
Please help me to understand why the unit ball of a separable banach space is metrizable, when it is given the induced topology from the weak topology. Specifically, my problem is I think I'd be able ...
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29 views
Let $(X,d)$ be a metric space and assume that $B_r^d(x)=B_s^d(y)$. is $r=s$ and $x=y$ [duplicate]
Let $(X,d)$ be a metric space and assume that $B_r^d(x)=B_s^d(y)$ where:
$$B_r^d=\{ a \in X | d(a,x) < r\}$$
Now, is it always true that
(a) $r=s$
(b) $x=y$
I made an elaborate argument on this ...
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36 views
Is the fine uniformity of a metric space, metric?
Let $d$ be a metric on the set $X$ and let $\mathcal T$ be it's topology and $\mathcal D$ be the finest uniformity on $X$ which induces $\mathcal T$.
Does $\mathcal D$ have a countable base?!
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1answer
39 views
Let $F$ be a subset of a metric space $(X, d)$ such that $\overline{F}$ is compact. Show that $F$ is totally bounded.
Let $F$ be a subset of a metric space $(X, d)$ such that $\overline{F}$ is compact. Show that $F$ is totally bounded.
how can I able to solve this.I have no idea.thanks for your help
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2answers
54 views
How to prove boundary of a subset is closed in $X$?
Suppose $A\subseteq X$. Prove that the boundary $\partial A$ of $A$ is closed in $X$.
My knowledge:
$A^{\circ}$ is the interior
$A^{\circ}\subseteq A \subseteq \overline{A}\subseteq X$
My proof ...
4
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3answers
104 views
$f$ continuous and surjective, $d_1(a,b)\le d_2(f(a),f(b))$, $X$ complete implies $Y$ complete
Let $(X,d_1)$ and $(Y,d_2)$ be metric spaces. Let $f : X \to Y$ be continuous and surjective. Suppose $d_1(a,b)\le d_2(f(a),f(b))$ for all $a,b\in X$. How can we show that if $X$ is complete then $Y$ ...
2
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1answer
49 views
Is the mapping $ d : X\times X \mapsto \mathbb {R} $ continuous?
Where $ (X, d) $ is a metric space. I want to prove it using sequential criteria. How do I tackle it?
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2answers
35 views
If 2 open balls define the same space, is it true that x=y and r=s?
Let $(X,d)$ be a non-empty metric sapce, $r$ and $s$ are postive radii, and $b_r^{d}(x)=b^d_s(y)$ for some $x,y \in X$.
Is it true that $r=s$ ?
Is it true that $x=y$?
My answer would be ...
4
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2answers
35 views
A point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement.
Let $E$ be a subset of a metric space $(S,d)$.
Prove that:
A point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement.
Here is what I thought:
I'm ...
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2answers
64 views
Does the union of open neighborhoods of all points in a metric space cover the metric space?
Let $M$ be a metric space that is locally compact. Let $O_i \subset M$. Let $C$ be an open cover of $O_i$, and let $C'\subset C$. Define $U \subset O_i$ to be an open neighborhood of some $x\in O_i$ ...
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2answers
114 views
Is this function Lipschitz continuous?
Let $\mu \in \mathbb R^d$ be given. Is the function $f:\mathbb R^d \to \mathbb R^d$ defined as $f(x) := \exp(-\|x- \mu\|) (\mu - x)$ Lipschitz continuous?
More specifically, for any $x, y \in ...
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Is the mapping that takes a metric to the induced intrinsic metric a closure operator?
To abbreviate the expression, "it holds that," I will write "iht."
First a definition. Given a partially ordered set $(P,\geq)$, a closure operator on $P$ is a mapping $\mathrm{cl} : P \rightarrow P$ ...
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64 views
Can we extend any metric space to any larger set?
Let $(X,d)$ be metric space and $X\subset Y$. Can $d$ be extended to $Y^2$ so that $(Y,d)$ is a metric space?
Edit:
how about extending any $(\Bbb Z,d)$ to $(\Bbb R,d)$
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61 views
equivalence of compactness and countably compactness
Is there a way to prove that in metric spaces, compactness and countably compactness are equivalent, without using the Bolzano Weierstrass Property?
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2answers
60 views
find open balls $B_1,B_2,B_3,\ldots$ so: $U=\bigcup _{n\in \Bbb N} B_n$ , where $U=\{(x,y)\in \Bbb R^2 : y\gt x\}$
In the metric space $(\Bbb R^2,d_{\Bbb R^2})$: How can I find open balls $B_1,B_2,B_3,\ldots$ so:
$U=\bigcup _{n\in \Bbb N} B_n$, where:
$U=\{(x,y)\in \Bbb R^2 : y\gt x\}$.
and why ...
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59 views
Show the $\operatorname{int}(A)$ is open.
So we want to show that the interior of any set $A$ is open.
We will denote $\operatorname{int}(A)$ as the interior of $A$ which is the set of all interior points of $A$.
I know in order to prove ...
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2answers
58 views
Intersection and union of complete subsets of a metric space
Does anyone know the two following proofs?
(i) the intersection of any collection of complete subsets of metric space $(X, d)$ is complete.
and
(ii) the union of a finite number of complete ...
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1answer
71 views
Proving a set is closed in a metric space
Given the subset
$$
F=\left\{f \in C[0,1]: \int_0^1 f(t)dt=1\right\}
$$
show that $F$ is closed in $C[0,1]$ with the supremum metric.
Definitions we use:
Limit point: $x$ is a limit point ...
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55 views
$U \in \tau(\Bbb R,d_\Bbb R)$ $\iff$ there are open intervals $B_1,B_2,B_3,…$ with $U= \bigcup_{n\in\Bbb N} B_n$
The rational numbers are countable: you can write $\Bbb Q =${$q_1,q_2,q_3,...$}.
Moreover,$\Bbb Q$ is dense in $(\Bbb R,d_\Bbb R)$.
Use these facts to prove for a non-empty set ...
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2answers
79 views
Show that the set of all orthogonal matrix of order $n$, $O(n)$ is a compact subset of $GL(n,\mathbb R)$
I have only concept in topology, metric space, and functional analysis. How do I tackle this. Also I want to know that is the set connected?
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61 views
Define metric on set and products
Let $X$ be set. My question is: if adding point $\ast$ to $X$ to get set $X \cup \{\ast\}$ then on countable product $\prod_{n \in \mathbb N_+} X \cup \{\ast\}$ I found it possible to define metric. ...
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3answers
75 views
Grasping the definition of open and closed sets
In a metric subspace $S = [0,1]$ of $\mathbb{R}^1$, why is it that every interval of the form $[0,x)$ or $(x,1], x\in (0,1)$, is an open set in $S$?
I understand that if you were to remove either, ...
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1answer
82 views
Show that closed unit ball in $ l^2 $ is not compact. [duplicate]
I have a prove using the defination of compact set.But I want to prove it using sequential criteria of compactness.How is it possibe?
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3answers
138 views
A example of closed and bounded does not imply compactnesss in metric Space
Let $X$ be the integers with metric $ρ(m,n)=1$, except that $ρ(n,n)=0$. Check that $ρ$ is a metric. Show that $X$ is closed and bounded, but not compact.
This is a "made-up" example demonstrating ...
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1answer
50 views
How to show that Hausdorff distance is a metric on the set of all compact non-empty subsets of a Polish space?
For each perfect Polish space $X$, let $H[X]$ be the set of all compact non-empty
subsets of $X$. If $x ∈ X$ and $A ∈ H[X]$, put
$$d(x,A) = \inf \{d(x, y) : y ∈ A\}$$
where on the right $d$ ...
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50 views
What are norms used for?
These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
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1answer
94 views
Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous?
Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$.
Is $f$ continuous?
Let $f$ be ...
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1answer
40 views
What is the name for this relation between metric spaces?
Consider two metric spaces on the same set $(X,d_1)$ and $(X,d_2)$ such that for all $x,y,z$, we have
$$ d_1(x,y)\leq d_1(x,z) \Leftrightarrow d_2(x,y)\leq d_2(x,z) $$
Is there a certain name to ...
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1answer
72 views
Topologically equivalence of a metric on matrices
Define a function on the set of $n\times n$ matrices by $\rho(A,B)=\operatorname{rank}(A-B).$ Prove that $\rho$ is a metric that is topologically equivalent to the discrete metric.
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Find a number that minimizes distance to a vector of sets of numbers
Assumptions
$V$ is a vector of sets $V_1,V_2,...,V_n$ of numbers:
$V=[V_1, V_2,..., V_n]^T, \forall_{i=1..n}V_i\subset\mathbb{R}$
$c\in\mathbb{R}$ is constant
$d(V,c)$ is an error metric: ...
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3answers
160 views
The real line has cardinality at most $\aleph_2$, but transfinite ordinal space has arbitrarily high cardinality: what is wrong?
In the context of supertasks, people and mathematicians are comfortable with the idea of transfinite ordinal time, that is, that time can be divided into an arbitrarily high number of steps. In most ...
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2answers
131 views
If $f: M\to M$ an isometry, is $f$ bijective?
$f: M\to M$ an isometry between metric spaces, is $f$ bijective?
$f$ obviously is injective. I proved bijection for $M=\mathbb{R}^n$. But I'm not sure if is true in general metric spaces.
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1answer
71 views
why must a normed space homeomorphic to a complete metric space be complete?
Why must a normed space X homeomorphic to a complete metric space Y be complete?
I've solved this in the case where Y is a normed space (considering open balls gives Lipschitz equivalence), but am ...
2
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0answers
75 views
For a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$, if $F$ is compact on compact subsets of $X$, then $F$ is compact on $X$
The problem as stated in the title isn't quite correct. Let $X$ be a topological space. What I have is a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$ which on compact subsets ...
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1answer
53 views
Compactness, Local Compactness, Completeness
Clearly, every compact metric space is locally compact. I always get confused when completeness is introduced into the picture. Which of the following are true? What are some easy counterexamples to ...
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3answers
108 views
Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space?
I have question.
Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space ? Please can you give me an advice some book names?
Thank you!
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1answer
66 views
Why $f(x) = \frac{d(x,A)}{d(x,A)+d(x,B)}$ is uniform continuous?
Let $X$ be a metric space, $A$ and $B$ are two subsets of $X$. $d(x, A) = \inf_{z \in A}d(x,z)$ and $\inf_{x \in A,y \in B}d(x,y) = \delta > 0$ We define $$f(x) = \frac{d(x,A)}{d(x,A)+d(x,B)}$$
...
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1answer
30 views
Maximum volume change for two sets with small Hausdorff metric in bounded part of $\mathbb{R}^n$
Given two subsets $S_1$, $S_2$ of a bounded part of $\mathbb{R}^n$, say $[-M,M]^n$. Is there a way to relate the difference in volume $vol(S_2)-vol(S_1)$ to the Hausdorff metric distance between the ...
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2answers
50 views
A question about metric spaces
Assume that we have a metric space $(S,d)$ and points $a,b,c \in S$ which statisfy the following conditions:
for all $x \in S$, $d(a,x) \leq d(a,b)$,
for all $y\in S$, $d(b,y) \leq d(b,c)$.
Does ...
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3answers
76 views
Product of connected spaces
You have two connected topological spaces $(A,B)$. Prove that $A\times B$ is also connected.
I understand that I have to prove that there is a point in $B$ (call it $b$), that makes $A\times\{b\}$ ...
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1answer
25 views
Compact metric space: proof $\text{diam}(K)$
I am to assume that $K$ is a compact metric space. I must prove that there are two points $x,y$ contained in $K$ such that $d(x,y)=\text{diam}(K)$.
Recall $\text{diam}(K)= \sup \{ d(x,y) \mid x,y ...
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2answers
52 views
Open Close Sets On Metric Space Help
I am having a tough time, trying to understand $C([0,1])$ the sup metric space. I can even prove that it is complete. However, I encountered the following question and I really really have no idea ...
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2answers
216 views
Show that the infinite intersection of nested non-empty closed subsets of a compact space is not empty
I'm given the following problem:
Suppose that for every $n\in \mathbb{N}$ $V_n$ is a non-empty, closed subset of a compact space $X$, with $V_n \supseteq V_{n+1}$.
Now I have to show that ...
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1answer
68 views
For a given metric space, how to show that $d(f(x),x) \ge \epsilon$?
I've got a quick question again. Suppose that we have a compact metric space $X$ which has a corresponding metric $d$. Also we know that $f:X\rightarrow X$ is a continuous map, such that for every $x ...
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5answers
260 views
Show that unit circle is compact?
Quick question. Say we are given the unit circle $\{ (x,y)\in \mathbb{R}^2: x^2+y^2=1 \}$.
Is this set compact? How can I prove that this is closed? Bounded? Do I have to take the complement of the ...
2
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1answer
84 views
determine whether or not a subset is closed or open
determine whether or not a subset is closed or open:
(a) For $X=\Bbb R^2$ and $d$ the Euclidean metric on $\Bbb R^2$:
$A_1=${$(x,y): x^2+y^2 <1$} $\cup $ {$(1,0)$}.
$A_2=${$(x,0): 0 ...
1
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1answer
74 views
Trying to complete a proof for my real analysis course (total boundedness)
Sorry for the long title. I'm trying to prove that any totally bounded subset S of a metric space X contains finitely many points such that the union of the open epsilon balls centered at these points ...
2
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1answer
30 views
Correctness of Converging sequence and Adherent Points
$x\in X$ is an adherent point of $A\subset X$ if for every $\epsilon>0$ there exists $y\in A$ s.t. $y\in B(x, \epsilon)$
$B(x, \epsilon)$ is the open ball centered at $x$ with radius $\epsilon$
...
