# Tagged Questions

43 views

### Example of a locally compact metric space whose completion is not locally compact

Can someone suggest an example of a locally compact metric space whose completion is not locally compact?
56 views

### Metrizability of quotient spaces of metric spaces

Suppose $X$ a metric space and $\sim$ an equivalence relation on $X$. Is the space $X/\mathord{\sim}$ metrizable? I think that the answer is no, but I could not arrive at a counterexample.
39 views

### Discontinuous function whose restriction on closed sets is continuous

Let $X$ a metric space, $\{U_i\}$ a collection of non-empty closed sets whose union is all of $X$. Give an example of a function $f:X\rightarrow \mathbb{R}$ such that the restriction $f|_{U_i}$ is ...
65 views

### If $S \subseteq X$ is closed, is $f(S,r)$ necessarily closed?

Let $X$ denote a metric space. Whenever $S \subseteq X$ and $r \in \mathbb{R}_{\geq 0}$, write $f(S,r)$ for the following set. $$\{x \in X \mid \exists s \in S : d(x,s) \leq r\}$$ Question. ...
42 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 ...
27 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 ...
33 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 ...
955 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 ...
66 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. ...
30 views

142 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 $Gâ€Ž_{Î´}$â€Žâ€Žâ€Ž â€Ž-subsets of $X$ . For example, Every first countable $T_1$ ...
97 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 ...
472 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 ...
642 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 ...
76 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, ...
95 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 ...
976 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 ...
465 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 ...
674 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 ...
### 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 ...