0
votes
2answers
28 views

Proving no finite basis of the system of neighborhoods at $a$ in the real line exist.

I'm not sure how to prove it, the gist is: I need to find the "smallest" neighborhood in the basis, take a ball of half that radius and show "look, there is no member of the basis in this ball, thus ...
0
votes
1answer
37 views

Show that for $(X,d)$ a metric space, $U= \{x \in X: d(x, C) \leq d(p, C)\}$ is a closed set

Let $(X,d)$ be a metric space, $C$ be a closed set in $X$. Define $$d(C, x) := \inf \{d(c, x): c\in C \}$$ for all $x \in X$. Fix a point $p \in X$. Show $U= \{x \in X: d(x, C)\leq d(p, C)\}$ is a ...
3
votes
0answers
49 views

Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
1
vote
1answer
91 views

Number of subsets/open subsets/closed subsets of a metric space.

Let $(X,d_1)$ and $(X,d_2)$ be two metric spaces which have the same infinite set $X$, but the different metrics $d_1$ and $d_2$. Denote the collection of subsets $X$ by $S$, and the collection of ...
0
votes
1answer
63 views

Continuity, and continuity in topology.

Metric spaces: Neighborhood of a point $a$ is a Set of point $N$, such that $\exists\delta>0:B_\delta(a)\subset N$ ($B_r(x)$ = open ball at x of radius r) Definition of open set: "A subset $O$ of ...
1
vote
1answer
39 views

Choice of Metric Gives Nice Topological Properties

I am looking for examples where choosing one possibility out of many for a metric gives nice topological properties compared to the other choices. Nice is defined as compact, Hausdorff, or whatever ...
1
vote
3answers
152 views

Let $X$ be a metric space with metric $d$. Show that $d:X \times X \longrightarrow \mathbb{R}$ is continuous.

Can someone please verify my proof or offer suggestions for improvement? I am aware that there is a similar question elsewhere, but I want help with my proof in particular. Let $X$ be a metric ...
1
vote
1answer
29 views

Homeomorphism of Interior of Convex Polygon to Open Unit Disk

Show that the interior of a non-degenerate convex polygon in $\mathbb{R}^2$ is homeomorphic to the open unit disk in $\mathbb{R}^2$. My attempt: Let $P$ be the set of points in the polygon, let ...
2
votes
0answers
30 views

Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
2
votes
1answer
21 views

Find continuous $f$ s.t. $f(x,y)^2 = x^2$ ($f: H \to \mathbb{R}$, H connected)

(i) Show that a metric space M is connected if and only if every continuous integer-valued function on M is constant. (ii) Show that $H = \{(x, y) \in R^2 : x > 0\}$ is connected. By ...
1
vote
1answer
33 views

Prove that square metric on $\mathbb{R}$ is in fact a metric.

In Munkres's topology, he proves that square metric on $\mathbb{R}^n$ is in fact a metric. By the square metric, I mean this function: $P:\mathbb{R}^n\times \mathbb{R}^n \rightarrow \mathbb{R}$ ...
1
vote
0answers
15 views

Easier proof of “countable hypocompactness”

I am interested in the following result, which appears as an old qual problem: Let $X$ be a metric space and $\{U_i\}$ a countable open cover. Prove that there exists a countable open refinement ...
4
votes
3answers
39 views

Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
0
votes
0answers
29 views

Question about Boundary points of the sets in metric space

Let A be a metric spaces. Prove the following properties: The boundary of $A$ equals $A'-A$ The boundary of $A$ is the closed set. $A$ is closed if and only if it contains its boundary. Where ...
3
votes
2answers
74 views

Riemannian manifolds are metrizable?

I've seen lots of casual claims that Riemannian manifolds (even without assuming second-countability) are metrizable. In the path-connected case, we can use arc-length to create a distance function. ...
6
votes
3answers
291 views

Given an example of a metric space in which every sphere has two centers

This is a question from Wilansky "Topology for analysis", P.15 Prob. 103 Maybe I was thinking too Euclidean, I can't come up some other "centers" of the sphere :(
2
votes
4answers
189 views

The set of points where two continuous functions agree is closed.

I want to prove that if $f,g$ are continuous functions from a topological space $(X,\tau)$ to a metric space $(Y,d)$ then the set $$ A = \{ x \in X : f(x) = g(x) \} $$ is closed. I found a very ...
3
votes
0answers
21 views

Ultrametric space of stochastic filtration

Let $\Omega$ be an arbitrary set and $(\mathscr F_t)_{t\in \Bbb R_+}$ be a non-decresing sequence of $\sigma$-algebras on $\Omega$ such that any subset of $\Omega$ is contained is some of them, that ...
8
votes
2answers
120 views

A metric on $\mathbb{N}$

Define a metric on $\mathbb{N}$ by fixing a prime, $p$, and setting $$d(x,y)=\begin{cases} 0 & x=y \\ p^{-k} & \text{otherwise} \end{cases}$$ where $p^k$ is the highest power of $p$ that ...
1
vote
0answers
18 views

Completeness of Locally Compact Metric Space and Group of Isometries

Let $X$ be a locally compact metric space, and suppose that the group of isometries of $X$ acts transitively. Show that $X$ is complete. (This is 2nd part of a problem. In first part I showed that for ...
2
votes
1answer
70 views

d' is finer than d on compact space $\implies$ $\forall \epsilon \ \ \exists \epsilon' \ \ \forall x \ B_{d'}(x,\epsilon') \subseteq B_d(x,\epsilon)$

This is my conjecture, but I guess I am missing the key idea for the proof (or my conjecture is wrong) Let d and d' be two metrics on a compact space $X$ ($X$ is compact with respect to both ...
2
votes
1answer
61 views

Definition of disc and open ball

I have the following definitions in my notes for arbitrary discs and open balls - $$D^n = \{x \in \mathbb{R^{n+1}}: ||x|| \le 1\}$$ $$B^n = \{x \in \mathbb{R^{n+1}}: ||x|| < 1\}$$ The ...
1
vote
2answers
49 views

Show that for any subsets $A,B\subset X$: (i):$d(A\bigcup B)\leq d(A)+d(B)+d(A,B)$ and (ii) $d(\bar A)=d(A)$

Let $d$ be a metric on $X$. Show that for any subsets $A,B\subset X$: (i) $d(A\cup B)\leq d(A)+d(B)+d(A,B)$ (ii) $d(\bar A)=d(A)$ I found this hard to prove because the diameter ...
3
votes
1answer
42 views

Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?

Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon >0$: $$ U_\varepsilon=\{x\in U: dist (x, \partial U)> \varepsilon \}. $$ Then $U_\varepsilon$ is open but in ...
1
vote
1answer
49 views

prove that there is maximum and minimum for C[0,1]

let $C[0,1]$ be the set of continuous functions $f:[0,1]\rightarrow\mathbb R$ and let $A\subset C[0,1]$ be the sub set of twice differentiable functions in $C[0,1]$ which satisfies: $\forall f\in ...
0
votes
0answers
51 views

Interior of a Dirichlet domain in a Riemannian manifold

Let $X\neq\varnothing$ be a complete connected Riemannian manifold. Suppose $G$ is a group of isometries of $X$, acting properly discontinuously on $X$. We assume there is a point $x_0\in X$ such that ...
-1
votes
0answers
31 views

Separable metrics

Is the following true?: Let $d_1$ and $d_2$ two separable metrics in space $X$. Then $d=\max(d_1,d_2)$ is a separable metric on $X$. Thanks!
0
votes
0answers
16 views

Proof verification related to the discrete metric

Can someone please verify my proof? Let $X_1$ be a set and let $d_1$ be the discrete metric on $X_1$. (a) Prove that every subset of $(X_1, d_1)$ is open. (b) Prove that if $(X_2, d_2)$ ...
1
vote
0answers
26 views

The future of the orbit of a point is a closed set [duplicate]

$X$ is a metric space and $f: X \rightarrow X$ is a dynamical system. Prove: $w(x_{0})$ is closed. Here the set $w(x_{0})$ is the future of the orbit of $x_0$, defined as $$\omega(x_0) = \{y \mid ...
1
vote
1answer
71 views

What is the metric spaces needed to motivate concepts of general topology?

I intend to start learning some topology on my own. I wonder How much metric spaces I should know in order to motivate the concepts of topology? I know it's possible to learn topology without any ...
2
votes
1answer
53 views

For compact $K$ and open $U \supseteq K$, there exists $\varepsilon>0$ such that $B(K,\varepsilon) \subseteq U$

Let $X$ be a metric space. Let $K$ be a compact subset of $X$ and $U$ an open subset of $X$ containing $K$. I strongly believe and want to prove that there exists $\varepsilon>0$ such that ...
3
votes
1answer
61 views

Qualifier problem: Completeness of Metric Spaces

I am working on old qualifier problems as a review, and I came across this one: Suppose there exists a continuous surjection $f:X_1 \mapsto X_2$, where $(X_1,d_1),(X_2,d_2)$ are metric spaces, such ...
4
votes
0answers
106 views

Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have ...
1
vote
0answers
85 views

Has anyone seen this space before? Does it have a name?

See the space below (the set taken as a subspace of the plane). It sort of looks like a comb, but with the wrap-around portion added, and the lower left corner removed. What would be a good name ...
0
votes
1answer
37 views

Characterization of closed sets in metric spaces

Let $X$ and be $Y$ be two metric spaces and $f:X\to Y$ a continuous function. We know that if $A$ is a closed set in $Y$ then $f^{-1}(A)=\{x\in X, \ \ f(x)\in A \}$ is a closed set in $X$. Now if we ...
0
votes
1answer
76 views

If image of closure belongs to closure of image, how to show preimage of interior belongs to interior of the preimage?

Here is exactly what I mean: Define a function $f:X\rightarrow Y$ from a metric space $X$ to another metric space $Y$. If any subset $A$ of $X$ satisfies $f(\bar A)\subset \overline {f(A)}$, then for ...
3
votes
0answers
26 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. ...
2
votes
0answers
33 views

Connected $G_\delta$ non-singleton, proper subsets in a connected complete metric space with more than one point

This is a question related to my last; I have still not solved it. Maybe this one is easier: Suppose $X$ is a connected complete metric space with more than one point. Must $X$ contain a ...
4
votes
0answers
60 views

Connected open proper subsets of a connected complete metric space

Suppose $X$ is a connected complete metric space with more than one point. Must $X$ contain a non-singleton non-empty connected proper open subset?
0
votes
0answers
34 views

Order of refinement of an open covering of $X$, a metric space

If every finite open covering of a metric space $X$ has a refinement of order $\leqslant n$, is it true that every open covering does too? We say that a covering has order $n$ if $n$ is the largest ...
0
votes
0answers
26 views

The number of non-degenerate proper subcontinua in a non-degenerate continuum

A continuum is any compact connected metric space. A continuum is non-degenerate if it is not a single point. My question is thus this: How many non-degenerate proper subcontinua must a ...
2
votes
1answer
58 views

Proving the metric attains a minimum on a compact subset

Let $(X,d)$ be a complete metric space. Suppose $B \subset X$ is compact. Prove that for every $a\in X$ the minimum $\min_{b\in B} d(a,b)$ exists. I'm pretty sure you can do this by just using the ...
1
vote
1answer
49 views

Connected $G_\delta$ sets in a connected completely metrizable space with more than one point.

Suppose $(X,\tau)$ is a connected completely metrizable space with more than one point. Let $\mathbb{G}$ be the set of all connected $G_\delta$ subsets of $X$. And let $\mathbb{O}$ be the class of ...
0
votes
2answers
58 views

In a metric space a compact set is closed

I want to show the following: Let $X$ be a metric space. Show that every compact subset $Y$ of $X$ is closed. The idea is to show that $X\setminus Y$ is open. So, for any $x \in X\setminus Y$, I ...
4
votes
1answer
60 views

Do connected complete metric spaces always contain a path?

Does every connected complete metric space with more than one point contain a non-trivial path? The pseudo-arc is an example of a connected metrizable space without a path.
3
votes
1answer
34 views

Connected complete metric spaces with more than one point.

Does every connected complete metric space with more than one point have infinitely many closed balls? And is any closed ball in a connected complete metric space connected?
0
votes
1answer
43 views

What is the difference between a Metric Space and a Pseudo-Metric Space?

I was wondering if anyone had information that would help me better understand the difference, so that I can work better on: Interesting Metrics I took a look at Metric assuming the value infinity ...
2
votes
1answer
35 views

Showing that the minimum distance between a closed and compact set is attained

I have two subsets of $\mathbb{R}^n$, given by $K$ and $F$, $K$ is compact and $F$ is closed. I'm trying to show that $\inf\{ d(x,y) : x \in K, y \in F \}$ is attained. My ideas so far: I know that ...
2
votes
1answer
75 views

Isometries from $\mathbb{R}$ to itself.

Prove that every isometry from $\mathbb{R}$ to itself is either $j_a$ or $i \circ j_a$. Here, $j_a$ is defined as $x\mapsto x+a$, and $i$ is defined by $x\mapsto -x$. Also, we're assuming the ...
0
votes
1answer
27 views

Showing infimum of distance is attained

I have a continuous map $f: X \to X$ on a compact metric space and I am trying to show that $inf \{ d(x,f(x)) : x \in X \}$ is attained. My thoughts so far are to use sequential compactness to obtain ...