# Tagged Questions

Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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### 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?
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### What are the epis in Met?

I have an assignment to precisely describe epimorphisms and monomorphisms in Met (category whose objects are Metric spaces and whose morphisms are contractions). I have shown that Mono $\iff$ one-to-...
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### Point-set topology - derived sets, boundaries, etc.

Consider the metric space $(\mathbb R^2,\sigma)$, where $\sigma$ is the discrete metric. Let $$E=\lbrace(x,y)\in \mathbb R^2 : x^2+y^2<1 \rbrace$$ i.e. the unit disc. Find the following sets [note ...
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### $S^1$ with length metric is not isometric to any subset of Euclidean plane (metric given by restriction)

Let $S^1$ denote point whose radius is 1 from the center. Metric is given by distance between two point is the shortest distance, that is the length metric. Prove that $S^1$ with this metric is not ...
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### Doubts about definition of open sets in “Understanding Analysis” by Stephen Abbott

In the book "Understanding Analysis" by Stephen Abbott, the author defines an open set as: A set $O \subseteq \mathbb{R}$ is open if for all points $a \in O$ there exists an $\epsilon$-...
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### Compactness of a set of bounded functions in the uniform norm

Let $T$ be a non-degenerate compact interval in $\mathbb R$ and $f:\mathbb R^2\to\mathbb R$ a strictly concave function such that (a) $f(0,0)=0$, (b) $f$ strictly increases in the first argument, and (...
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### Give an example of A continuous function from X onto Y where X=[0,1] ;Y=[0,1]×[0,1] [duplicate]

Give an example of A continuous onto function $f:X\to Y$ where $X=[0,1]$ ;$Y=[0,1] \times[0,1]$.Why can't this function be one to one on [0,1]? As far as an example is concerned the only way to ...
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### What is $d(\sin(x),\cos(x))$ if d is a distance function in a metric space?

Let $M=\{f:[a,b] \to \textbf{R} | f \,is \,continuous \}$. Let $d:M \to \textbf{R}$ be defined by $d(f,g)=\int_a^b |f(x)-g(x)| \,dx$. What is d represent geometrically, and show that M, d is a metric ...
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### Point-set topology, characterizing sets

I think the question sort of speaks for itself. Please no solutions in the answers - I'm mostly looking to see if my logic makes sense: Let $E$ be a set in a metric space $(X,d)$. Let $E^{\circ}$ ...
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### Finite intersection property in any metric space

If $(X,d)$ is any metric space and $\{A_\alpha\}_{\alpha\in I}$ is a collection of nonempty compact subsets of $X$ such that the intersection of any finite subcollection of sets is non empty does that ...
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### Does uniform convergence depends on the metric?

Definition: Let $f_n:X\to Y$ be a sequence of functions from a set $X$ to the metric space $Y$. Let $d$ be the metric for $Y$. The sequence $(f_n)$ d-converges uniformly to the function $f:X\to Y$ if ...
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### Assumptions of Kolmogorov extension theorem

As far as I know, a class of spaces for which the Kolmogorov theorem works and which is closed under countable products, are the spaces of complete separable metric spaces which are also called Polish ...
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### The metric identification of a pseudometric on $C(\mathbb{I})$
I have a pseudometric $\mu$ on $C(\mathbb{I})$ defined by $$\mu(f, g) = |f(x_0) - g(x_0)|.$$ I then take the metric identification of $(M, \mu)$ and am asked what familiar space this metric ...