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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Prove that $C = \{x \in X: (f_n(x)) \text{ converges} \}$ is a closed set in $X$

The question goes like: Let $X$ be a compact metric space and let ${f_n}$ be an equicontinuous sequence in $C(X)$. Show that $C = \{x \in X: (f_n(x)) \text{ converges} \}$ is a closed set in $X$ My ...
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A Compact Hausdorff space which is locally metrizable is metrizable.

This is exercise 7 from section 34 in Munkres. The hint given is to show that the space is a union of finitely many subspaces which are second countable. This question has been asked before A ...
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32 views

Closure operation is not necessarily preserved [closed]

If $(X,d)$ and $(Y,d')$ are metric spaces and if $f:X\rightarrow Y$, show that $f$ is continuous if and only if for every $A\subset X$, $f(\overline{A})\subset\overline{f(A)}$, and construct ...
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73 views

Topology: Difference between Bolzano-Weierstrass Property and Sequential Compactness?

In a general topological space are these properties equivalent ? If not, is there a property (e.g. first countability) that metric spaces possess which makes them equivalent there ? Here are the ...
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35 views

Completeness of metric space defined by a continuous bijection?

Suppose $f: X \to Y$ is a continuous bijection.Given a metric $d$ on $X$ we can define a metric $e$ on $Y$ as: $e(y_1,y_2)= d(f^{-1}(y_1),f^{-}(y_2))$. $1$. Suppose $(X,d)$ is complete,does this ...
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$X$ be a set such that any two metric on $X$ are equivalent ; then is it true that $X$ has to be countable? Can $X$ be countably infinite?

Let $X$ be a set such that any two metric on $X$ are equivalent (i.e., generates the same topology i.e. any metric on the set generates discrete topology); then is it true that $X$ has to be ...
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Does there exist a $f: \bf R \to \bf R$ which is differentiable,uniformly continuous and lim$_ {x \to \infty} f'(x)=\infty$?

Does there exist a $f: \bf R \to \bf R$ which is differentiable,uniformly continuous and lim$_ {x \to \infty} f'(x)=\infty$ ? I'm unable to find a counterexample.I know that if derivative is bounded ...
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36 views

Pseudometric from nested sets

Assume that $X$ is a nonempty set and that for every $x\in X$ we have a nested family of nonempty sets: $\{E_{x,t}: t\in [0,\infty)\}$ for which $t_1\leq t_2$ implies $E_{x,t_1}\subseteq E_{x,t_2}$, ...
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Show that $d(x,y)=\dfrac{1}{k(x,y)}$ is metric .

Let $X$ be a non empty set. Let $M$ the set of all sequences $(x_{n})$ of elements of $X$. For $x=(x_{n})$ and $y=(y_{n})$ in $M$, let $k(x,y)$ the smallest integer $n$ such that $x_{n}\neq y_{n}$. ...
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Is $C[0,1] \setminus P$ (where $P$ is the set of polynomials) connected?

Here's my latest try: Suppose it's not. It means there are two non-empty disjoint open sets $A,B$ such that $A \cup B = C[0,1] \setminus P$. "Take" $f \in A, g \in B$ such that $f(x)<g(x) (\forall ...
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138 views

Prove that a function is continuous in a metric space

Here is the problem : Let $(X,d)$ be a metric space, and let $A$ be a non-empty closed subset of $X$ $($$\varnothing\neq$$A$$\subset$$X)$ , and let $f:A\to\mathbb{R}$ be a bounded continuous function ...
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52 views

Pointwise boundedness of continuous functions on a complete metric space implies uniform boundedness

I am in the process of proving the following: Let $\mathcal{F}$ be a family of continuous real-valued functions on a complete metric space $X$ that is pointwise bounded in the sense that for each ...
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23 views

On extending a function $f:\mathbb N \to [-1,1]$ to $\bar f:\mathbb R \to [-1,1]$ such that $\bar f$ is differentiable at every point of $\mathbb N$?

For which type of functions $f:\mathbb N \to [-1,1]$ , can we extend it to a continuous function $\bar f:\mathbb R \to [-1,1]$ such that $\bar f$ is differentiable at every point of $\mathbb N$ ? ...
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Let $X \subset \mathbb{R}^n$. Suppose that $0 \in X$ and $\|x-y\| = 1$ for $x,y \in X, x \neq y$. Then the maximum number of elements in $X$ is $n+1$.

Let $X \subset \mathbb{R}^n$. Suppose that $0 \in X$ and $\|x-y\| = 1$ for $x,y \in X, x \neq y$. Then the maximum number of elements in $X$ is $n+1$. My attempt By contradiction, let's suppose that ...
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50 views

What values of $p$ give convergence to $0$ in $l^p$

Given a sequence $x_n \in l^p$ whose first $n^2$ members equal $\frac {1}{n}$, and all other entries $=0$, for what values of $p$ does the sequence converge to the zero sequence in $l^p$? So do I ...
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29 views

Disjoint closed sets in a metric space [duplicate]

Give an example of a metric space $(X,d)$ and two closed sets $A,B$ with $A\cap B=\varnothing$ and $d(A,B)=0$.
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Stating if a function being a contraction

I have to state if a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is a contraction on the interval $I\subset \mathbb{R}$ and say if it admits fixed points. The function is $f(x)=\left\{ ...
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Extension of a uniformly continuous function, quibble

I stumbled over this question which is slightly different from the usual version I remember, e.g. here: in one case one extends the uniformly continuous from a metric space (not necessarily complete) ...
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67 views

Show a continuous function on a closed bounded interval is Lipschitz under the maximum (infinity) norm

I'm currently working on the following: Define the function $\psi: C[a,b] \to \mathbb{R}$ by $\begin{equation*} \psi(f)=\int_{a}^{b}f(x)\,dx \end{equation*}$ for each $f \in C[a,b]$. Show that ...
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Prove that $f:\mathbb{R}\rightarrow S$, defined by $f(x)=s_x$ is continuous

Let $S\subset \mathbb{R}$ a closed set such that for all $x\in\mathbb{R}$ exists an only $s_x\in S$ such that $d(x,s_x)=d(x,S)$. Prove that $f:\mathbb{R}\rightarrow S$, defined by $f(x)=s_x$ is ...
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boundedness imply total boundedness

I have a question, we can consider it for metric spaces specifically. I wanted to ask if is there any known property a metric space $X$ may posses such that any closed bounded set will imply it is ...
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19 views

Convergence of intervals in the sense of Hausdorff distance

For the definition of Hausdorff distance, please see here Suppose I have a sequence of interval $I_n=[a_n,b_n]\subset [0,1]$, then I read in a paper it says that, up to a subsequence, $I_n\to I$ in ...
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26 views

Showing sets are open in a metric space.

Let $d$ be the usual Euclidean metric on $\Bbb R$ and $f: (\Bbb R, d) \mapsto ([-1,1],d)$ be the map given by $f(x) = \left\{ \begin{matrix} 0, & x=0 \\ \sin \frac 1 x, & x \neq 0 \end{matrix} ...
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Convergence of sequence with one to one function

We are working in some metric space $(X,d)$. I am given that $\{x_n\}_{n=1}^\infty$ converges to $x$ and we suppose that $f: \mathbb{N} \rightarrow \mathbb{N}$ is a one to one function. We want to ...
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$D$ dense in $M$, $\{f_n\}$ sequence of equicontinuous maps, $f_n \rightarrow f$ pontwise in D. So $f_n \rightarrow f$ uniformly in $K$ compact

Let $M$ and $N$ be arbitrary metric spaces and $D \subset M$ dense. Given an equicontinuous sequence of maps $f_n\colon M \rightarrow N$ and a continuous map $f\colon M \rightarrow N$, suppose that ...
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Vector space containing vectors of infinite norm not complete?

Let $V$ be a vector space such that there is a $v \in V$ with $\|v\|_V = \infty$. Can you conclude from this that $V$ is not complete, i.e. that there is a Cauchy sequence in $V$ which does not ...
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For uniformly continuous $f:X\to Y$, where $X$ is a totally bounded metric space, prove that $f(X)$ is totally bounded

I am faced with the following problem: Let $X$ be a totally bounded metric space. If $f$ is a uniformly continuous mapping from $X$ to a metric space $Y$, show that $f(X)$ is totally bounded. Is ...
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Equivalence of uniform convergence in metric spaces!

Let $f, f_1, f_2, \dots, f_n, \dots$ be continuous applications $f_i: M \rightarrow N$, $ f: M \rightarrow N$. Then, the following affirmations are equivalent: $(1)$ If $x_n \rightarrow x$ in M, then ...
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$x_n \rightarrow x \implies \lim_{n \rightarrow +\infty}$ IFF $f_n \rightarrow f$ uniformly in each $K \subset M$ compact

Let $f, f_1, f_2, \dots$ be continuous maps from M to N, M and N metric spaces. The, the following affirmations are equivalent: $(a)$$ x_n \rightarrow x \implies \lim_{n \rightarrow +\infty} f_n(x_n) ...
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21 views

M compact. $(x_n)$ converges if and only if $(x_n)$ has only one closure point [duplicate]

M compact. $(x_n)$ converges if and only if $(x_n)$ has only one closure point. Show, with one example, that the compacity is necessary. My attempt: 1) Of course if $(x_n)$ converges, $(x_n)$ has ...
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Is there a metric space $X$ having either $1012$ , or $1036$ , or $1089$ many open sets?

Let $X$ be a metric space , $n$ be the no. of open sets in $X$ , then the possible values of $n$ are : 1) $1012$ 2)$1024$ 3)$1036$ 4)$1089$ I know that $1024$ is a possible value because it is a ...
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56 views

Which spaces are relatively compact and connected?

Let $X=\{(x,y):x^2+y^2<5\}$ and $K=\{(x,y) :1\leq x^2+y^2\leq 2\text{ or }3\leq x^2+y^2\leq 4\}$. Then which are true: $1$.$X\setminus K$ has $3$ connected components. $2$.$X\setminus K$ has no ...
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Metric space properties

I am doing this question from mainly an analysis viewpoint since I haven't covered metric spaces yet. I have completed part (i). Now I am stuck on (ii). How do I show it is well defined and ...
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48 views

Difference between $\mathbb{R}^{4}$ and $\mathbb{R}^{1,3}$

What's the difference between $\Bbb R^{4}$ and $\Bbb R^{1,3}$? I know that the first one has metric Kronecker delta $\delta_{ij}$. Does the second one have Minkowski metric $g_{\mu \nu}$?
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52 views

Separately continuous function vanishing on a dense set

This is a follow-up to If a separately continuous function $f : [0,1]^2 \to \mathbb{R}$ vanishes on a dense set, must it vanish on the whole set?. Is there an example of: a pair of metric spaces ...
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Is there a name for this property (on sequences)

Suppose we have a sequence $(x_n)_{n=1}^\infty\subseteq X$ with $(X,d)$ a metric space, and we have the following property: $$\exists x\in X\forall\epsilon>0[|B_\epsilon(x)\cap ...
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let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and $A \subset \mathbb{R}$

let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and $A \subset \mathbb{R}$ be defined by $A=\{y \in \mathbb{R} : y = \lim_{n\to\infty}f(x_n)$, for some sequence $x_n \to +\infty$}$.$ Then ...
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If $f$ takes Cauchy sequence to Cauchy sequence then $f$ is continuous [duplicate]

If $f:X\to Y$ takes Cauchy sequence to Cauchy sequence then prove that $f$ is a continuous function. Let $x_n$ be a sequence in $X$ such that $x_n\to x\implies x_n$ is Cauchy $\implies f(x_n)$ is ...
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If $f$ is a map from a topological space $Y$ to a metric space $X$, to prove that $f$ i

If $f$ is a map from a topological space $Y$ to a metric space $X$, to prove that $f$ is continuous at y, is it enough to show that for all $\epsilon >0$, there exists $V_{y}$ (neigbhourhood of $y$ ...
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35 views

Finite dimensional normed space

What is the largest possible $c$ in the equation shown in the following image for if $X = \mathbb{R}^2$ and $x_1 = (1, 0),x_2 = (0, 1)?$ If $X = \mathbb{R}^3$ and $x_1 = (1, 0, 0), x_2 = (0, 1,0), x_3 ...
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Completion of an Infinite Dimensional vector space

In the following I will write $\mathbb{R}^{\infty}$ and $c_{00}$ to mean the following sets: $\mathbb{R}^{\infty} = \left\{ \ x = ( x^{(1)}, x^{(2)}, x^{(3)}, \ldots, x^{(k)}, \ldots ) \mid x^{(k)} ...
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Let $(X,d)$ be a complete metric space. If $X$ has no isolated points then $X$ is uncountable.

Let $(X,d)$ be a complete metric space. If $X$ has no isolated points then $X$ is uncountable. I know that for a Hausdorff and compact general space it is true but, how do I use Baire category in ...
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Subspace of metrizable space is metrizable.

If X is a metric space I can consider the induced topology on x from the metric. Now I consider a subset of X, by restriction the metric on X induces a subspace metric on Y. So to prove that Y is ...
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How to 'get rid of' limit so I can finish proof?

Suppose $\sup_{x \in \mathbb{R}} f'(x) \le M$. I am trying to show that this is true if and only if $$\frac{f(x) - f(y)}{x - y} \le M$$ for all $x, y \in \mathbb{R}$. Proof $\text{sup}_{x \in ...
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Convergent Sequences and Open Sets

I have a question about convergent sequences in metric spaces: Let $(M,d)$ be a metric space. Show that $A\subseteq M$ is open if and only if there is no sequence $\{x_n\}_1^\infty$ in $M-A$ that ...
2
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1answer
42 views

$G_\delta$ set of nowhere differentiable functions?

I'm trying to show that in $C([0,1])$ with the supremum metric, there exists a dense $G_\delta$ set of nowhere differentiable functions. Honestly, I don't know how to approach this. Any help would be ...
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Terminology and Notation: “Sphere” in a Metric Sapce

I see sphere used interchangeably with "ball" in the sense of "open ball", "closed ball", with notation like $B_r(x), B_r[x]$. I also see sphere used to refer to the boundary of a closed ball, e.g. ...
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Problem involving the completion of a general metric space

A completion of a metric space $(X,d)$ is a metric space $( \widetilde{X}, \widetilde{d} )$ such that $X \subseteq \widetilde{X}$ and such that: (I) $\widetilde{d}$ extends $d$. That is, we have ...
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1answer
73 views

Set of all differentiable functions on $(0,1)$ not complete

Let $\mathcal{D}$ be the subspace of $C[0,1]$ (with uniform metric) consisting of the continuous functions $[0,1]\to \mathbb{R}$ that are differentiable on $(0,1)$. Is $\mathcal{D}$ complete? I'm ...
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52 views

Metric is continuous function

Let $X$ be a metric space , $d$ is the metric , show that $d$ is a continuous function from $X\times X$ to $R$. I think the definition is all we need , but I just don't know where to start , can ...