-1
votes
1answer
23 views

Some Topological Properties of Starlike Sets!

A subset $E$ of $\mathbb R^n$ is starlike if it contains a point $p_0$ (called a center for $E$) such that for each $q\in E$, the segment between $p_0$ and $q$ lies in $E$. For more information please ...
0
votes
2answers
32 views

Lipschitz does not imply fixed points

I have the following problem in mind: Let us say we have a function $f:X\rightarrow X$ (X is a complete metric space) and it respects that if $x\neq y$ then : $d(f(x),f(y))<d(x,y)$. My trouble ...
4
votes
4answers
919 views

Example of two open balls such that the one with the smaller radius contains the one with the larger radius.

Example of two open balls such that the one with the smaller radius contains the one with the larger radius. I cannot find a metric space in which this is true. Looking for hints in the right ...
0
votes
1answer
60 views

A counterexample on compactness (closed vs complete)

In a metric space $M$: If $A \subset M$ is complete and for each $\epsilon > 0$ there exists a compact $K \subset M$ with $A \subset \{ x \in M : d_M(x, K) \leq \epsilon \}$ then $A$ is compact. ...
2
votes
2answers
55 views

Continuous bijection whose inverse is not continuous at uncountably many points

I am interested in understanding to what extent continuous bijections fail to be homeomorphisms. For example, suppose $X, Y$ are metric spaces and $f: X\to Y$ is a continuous bijection. Is it possible ...
0
votes
1answer
96 views

Isometry of a metric space with proper subset

In Irving Kaplansky's "Set Theory and Metric Spaces", exercise 17 on page 71 asks for an example of a metric space which is isometric to a proper subset of itself. Any infinite discrete space and any ...
0
votes
1answer
58 views

Counterexample to: if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ with $\mu(X)=\infty$

We know if $\mu(X)<\infty$, and if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ (can be proved by using Holder's inequality). Is this still true if $\mu(X)=\infty$? Counterexample? ...
3
votes
1answer
412 views

Separable metric space has a countable base

A collection $\{V_{\alpha}\}$ of open subsets of $X$ is said to be a $\textit{base}$ for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have ...
0
votes
1answer
227 views

Examples of homeomorphisms between the real numbers and the positive real numbers?

I'm interested in homeomorphisms between the real numbers, $\mathbb{R}$, and the positive real numbers, $(0,\infty)$--where both spaces have the topology induced by the metric $d(x,y)=|x-y|$. Here ...
1
vote
3answers
1k views

Examples of metric spaces which are not normed linear spaces?

Give an example of a metric space which is not a normed linear space. Justify your example.
8
votes
3answers
117 views

Is this kind of space metrizable?

It has a nice result from Tkachuk V V. Spaces that are projective with respect to classes of mappings[J]. Trans. Moscow Math. Soc, 1988, 50: 139-156. If the closure of every discrete subset of a ...
4
votes
1answer
87 views

A question on metrizable space

This exercise is from "General Topology" by Engelking: Give an example of a metrizable space which cannot be embedded in a locally compact metrizable space. I don't how to start. The ...
3
votes
2answers
94 views

Metric spaces and distance functions.

I need to provide an example of a space of points X and a distance function d, such that the following properties hold: X has a countable dense subset X is uncountably infinite and has only one ...
2
votes
1answer
348 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 ...
6
votes
2answers
115 views

non compact nested closed sets in metric spaces.

Do you have an example for closed sets $...\subseteq F_4\subseteq F_3\subseteq F_2\subseteq F_1$ such that: $$\bigcap_{n=1}^\infty F_n=\emptyset $$ in $\mathbb{R}^n$ or a metric space?
3
votes
0answers
136 views

Moscow space-Examples

A space $X$ is called $Moscow$, if for each open subset $U$ of $X$, the closure of $U$ in $X$ is the union of a family of $G‎_{δ}$‎‎‎ ‎-subsets of $X$ . For example, Every first countable $T_1$ ...
3
votes
1answer
90 views

What are the use cases related to cluster analysis of different distance metrics?

I'm trying to use different distance metrics like Euclidean, Manhattan, cosine, chebyshev among other distance metrics in my k-means algorithm to calculate distances between the data points and the ...
19
votes
2answers
459 views

What is a metric for $\mathbb Q$ in the lower limit topology?

A useful source of counterexamples in topology is $\mathbb R_\ell$, the set $\mathbb R$ together with the lower limit topology generated by half-open intervals of the form $[a,b)$. For example this ...
2
votes
1answer
601 views

Intersection of countable set of compact sets

I am asking whether a specific construction is a counterexample to Theorem 2.36 in Rudin's "Principles..." book (3rd Ed.), which reads, 2.36 Theorem If $\{K_{\alpha}\}$ is a collection of compact ...
2
votes
1answer
75 views

A question about weakening the conditions of Schauder's fixed point theorem

I'm currently doing a course on the theory of metric spaces. This is the version of Schauder fixed point theorem from my course: Let $(X,\|\cdot\|)$ be Banach and $C\subset X$ a closed, bounded, ...
1
vote
0answers
94 views

Distinct metrics on a manifold

I'm trying to understand basic differential geometry (my background is in mathematical logic), and I'm having a bit of difficulty with a basic point. Frequently we want to consider the set of metrics ...
5
votes
1answer
968 views

Intersection of compact sets

I have a brief question about Theorem 2.36 in Baby Rudin. The theorem is as follows: If $\{K_\alpha\}$ is a collection of compact subsets of a metric space $X$ such that the intersection of every ...
6
votes
1answer
457 views

Sum of Cauchy Sequences Cauchy?

Let $(X,+)$ be an abelian group and $d$ a metric on $X$. Suppose $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences. What conditions on the relation between the group operation and the metric are sufficient ...
5
votes
2answers
648 views

Opposite of a contraction mapping

I am taking Real Analysis and we recently went over the Banach Fixed-point Theorem, also commonly known as the Contraction Mapping Theorem which states: If $(X,d)$ is a complete metric space, and ...
2
votes
1answer
174 views

Give an example that $\overline{A \cap B} \neq \overline{A} \cap \overline{B}$ [duplicate]

Possible Duplicate: Is the closure of $ X \cap Y$ equal to $\bar{X} \cap \bar{Y}$? I'm sorry to ask another question so soon after my last one, but my exam Introduction to Functional ...
3
votes
2answers
887 views

Contraction mapping does not hold in metric space

Let $X=\mathbb{Q}\cap [1,2]$, i.e $X$ is the set of rational number between 1 and 2 inclusive. We can consider $X$ to be a metric space by endowing it with the usual distance function, i.e for $x,y ...
4
votes
3answers
370 views

Is total boundedness a topological property?

If a metrizable topological space is totally bounded with one metric, is it totally bounded with all others? A related, stronger question: if every metrization of a topological space is bounded, are ...
11
votes
6answers
2k views

Examples of non symmetric distances

It is well known that the symmetric property is $d(x,y)=d(y,x)$ is not necessary in the definition of distance if the triangle inequality is carefully stated. On the other hand there are examples of ...