Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Can we prove, without invoking invariance of domain, that $\mathbb R$ and $\mathbb R^2$ are not homeomorphic?

Can we prove, without invoking invariance of domain, that $\mathbb R$ and $\mathbb R^2$ are not homeomorphic, or equivalently, that no open set of $\mathbb R$ is homeomorphic to an open set of ...
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Existence of continuous bijective function $f:[0,1] \times [0,1] \to [0,1] $ ? Continuous and only injective and continuous and olny surjective?

Does there exist any continuous bijective function $f:[0,1] \times [0,1] \to [0,1] $ , where $[0,1]$ is equipped with usual Euclidean metric of $\mathbb R$ and $[0,1] \times [0,1]$ is equipped with ...
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Discrete subset of $\mathbb R^2$ such that $\mathbb R^2\setminus S$ is path connected.

Let, $S\subset \mathbb R^2$ be defined by $$S=\left\{\left(m+\frac{1}{2^{|p|}},n+\frac{1}{2^{|q|}}\right):m,n,p,q\in \mathbb Z\right\}.$$ Then, which are correct? (A) $S$ is a discrete set. (B) ...
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How to prove d is metric if prime number is involved?

Please have a look at this question. Can anyone please tell me how to approach when dp(x,y) /= 0? Does this mean if x= y, then d=0, otherwise d = p^-k? or only if p^k satisfy the following ...
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Periodic Points of $h:=f\times g: [0,1]^2\to[0,1]^2$ for Continuous $f,g$

Question: Let $f,g:[0,1]\to[0,1]$ be continuous, $h:=f\times g:[0,1]^2\to[0,1]^2,$ $(a,b)\mapsto(f(a),g(b))$. Then "Period Three Implies Chaos" applies to $h$, while Sharkovskii's Theorem does not. ...
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To show that for each $\epsilon > 0$, there exists an infinite set $B \subset A$ such that diam$(B) < \epsilon$.

If $A$ is an infinite subset of a totally bounded metric space $(X, d)$, then to show that for each $\epsilon > 0$, there exists an infinite set $B \subset A$ such that diam$(B) < \epsilon$. ...
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Counterexamples for the Converse of “Topological Conjugacy Implies Equal Topological Entropy”

Question: I would like to find two topological dynamical systems that are not topologically conjugate but nevertheless have the same topological entropy. Two topological dynamical systems $f:X\to ...
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$U$ be open in $X$ and $A:=X \setminus U$ , is the function $f: U \to \mathbb R ; f(x):=dist (x,A) , \forall x \in U$ injective ?

Let $(X,d)$ be a metric space , $U$ be open in $X$ and $A:=X \setminus U$ , is the function $f: U \to \mathbb R$ defined by $f(x):=dist (x,A) , \forall x \in U$ injective ? If not , then do we need ...
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metric space is open and closed.

A open ball is a open set in metric. I know the complement of an open set is a closed set and union of an open set is open set. d(y, 1) = 1, d(y, 2) = 1, d(y, 3) = 1, d(y, 4) = 1 which is balls, ...
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To show that the set $A = \{ (x,y) : x,y \in \Bbb R, x \notin \Bbb Q$ or $y \notin \Bbb Q \} $ is neither open nor compact in $\Bbb R^2$.

To show that the set $A = \{ (x,y) : x,y \in \Bbb R, x \notin \Bbb Q$ or $y \notin \Bbb Q \} $ is neither open nor compact in $\Bbb R^2$. To show neither open nor compact in $\Bbb R^2$ is same as ...
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If $A_i$ is a compact subset of a metric space $(X_i,d)$ where $i = 1,2$ to show that $A_1 \times A_2$ is compact in $X_1 \times X_2$.

If $A_i$ is a compact subset of a metric space $(X_i,d)$ where $i = 1,2$ to show that $A_1 \times A_2$ is compact in $X_1 \times X_2$. Proof: Let $\{(a_n ,b_n)\}$ be any sequence in $A_1 \times A_2$ ...
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proof that if $(X,d)$ is a metric and $\mathbb{R}$ has the standard topology, with $C\subset X$ that the following map is continuous.

proof that if $(X,d)$ is a metric and $\mathbb{R}$ has the standard topology, with $C\subset X$ that the following map is continuous. $X \rightarrow \mathbb{R}$ by $x\rightarrow d(x,C)$ where ...
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To prove that $f(A)$ is compact in $(Y,e)$.

Let $(X,d)$ and $(Y,e)$ be metric spaces, $A \subset X$ is compact and $\eta$ a fixed number and $f : A \to Y$ a function such that $$e(f(x),f(y)) \leq \eta d(x,y) \ , \ \forall x,y \in A$$ To prove ...
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A metric space $(X,d)$ is compact if and only if it is bounded in every equivalent metric. [closed]

How can I prove that a metric space $(X,d)$ is compact if and only if it is bounded in every equivalent metric?
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$S$ is closed $\iff$ whenever $\{x_n\} \in S$ and $x_n \to x$, then $x \in S$

Definitions: Limit point: $x$ is a limit point of $S$ if $\forall r>0$, $\exists y \neq x$ such that $y \in B_r(x) \cap S$ Closed$_1$: A set $S$ is closed if its complement is open ...
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Question about metric spaces. [closed]

Prove, that for any infinite set $X$, there exists some metric $d$ such that $X$ has a limit point under $d$.
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$M_1$ is compact if and only if $M_2 = (A,d_2)$ is a complete metric space whenever $d_2$ is equivalent to $d_1$. [duplicate]

Let $M_1 =(A,d_1)$ be a metric space. Then $M_1$ is compact if and only if $M_2 = (A,d_2)$ is a complete metric space whenever $d_2$ is equivalent to $d_1$. I am finding difficulty in proving this ...
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27 views

To show that a subset $D$ of $X$ is dense iff it is $\epsilon$-net for every $\epsilon>0$.

To show that a subset $D$ of $X$ is dense in X if and only if it is $\epsilon$-net for every $\epsilon>0$. Let $D$ be dense in $X$ and $y \in X$, then there exists a sequence $(x_n)$ in $D$ ...
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When does continuous (or uniformly continuous ) function between normed linear spaces carries bounded sets to bounded sets ?

I know that if $f:\mathbb R^m \to \mathbb R^n$ is continuous then $f$ carries bounded sets to bounded sets . What if we say $X,Y$ are normed linear spaces and $f:X \to Y$ where $f$ is continuous ? ...
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Proving that Compact Implies Seperable in a Metric Space

Let $(X,d)$ be a Compact Metric Space. We wish to show that X is Seperable. Given $ \delta = 1/n \;$ , $\bigcup_{x\in X} D(x,1/n) \; $ is an Open Cover of X Since X is Compact $ \exists $ Finite ...
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coloring theorem for topological partitions

Let $(X,\tau)$ be a topological space. DEFINITIONS: Define a topological partition of $X$ into connected sets to be a collection of pairwise disjoint open connected sets $\{U_i\}_{i\in I}$ such that ...
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To show that $f$ has a fixed point, that is, there exists $x_0 \in X$ such that $f (x_0) = x_o$.

Let $(X, d)$ be a compact metric space. Let $f : X \to X$ be such that $d(f (x), f (y)) < d(x, y)$ for all $x, y \in X$ with $x \neq y$. To show that $f$ has a fixed point, that is, there exists ...
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35 views

Do homeomorphic metric spaces have equal minimal cardinality of dense subsets? [closed]

Let $X,Y$ be two homeomorphic topological spaces and let $d(X)$ denote the minimal cardinality of a subset $A \subseteq X$ such that $\bar A=X$, i.e., $A$ is dense in $X$. Then is it true that ...
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Let $f: X \to X$ be such that $d(f(x), f(y)) = d(x, y)$ for all $x, y \in X$. To show that $f$ is onto. [duplicate]

Let $(X, d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x), f(y)) = d(x, y)$ for all $x, y \in X$. To show that $f$ is onto. Since the function $f$ satisfies $d(f(x), f(y)) = d(x, ...
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Proving a set is closed in the space of continuous functions

(Question from Royden's Real Analysis) Let C be the space of all continuous real-valued functions on [0,1], equipped with the sup norm metric. Let $F_n=${$\exists x_0 \in [0,1]$ s.t. $\forall x \in ...
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Mahalanobis distance to the ellipsoid center

I am confused about the following description: If we parameterize the ellipsoid $E$ as: $E = \{x|\ ||Ax-b||_2 \leq 1\}$. $A \in S_{++}^n$ Then the Mahalanobis distance to the ellipsoid center is ...
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1answer
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Showing $\partial \partial S= \partial S$ for open sets $S$ in a metric space

Let $(X,d)$ be a Metric Space and $S \subset X$ an Open set Show that $\partial \partial S = \partial S$ I was wondering if my reasoning is right. We know $\partial S = Int \; \partial S \cup ...
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To show that there exists a non-empty subset $A$ of $X$ such that $f(A) =A$.

Let $X$ be a compact metric space. Let $f: X \to X$ be continuous. To show that there exists a non-empty subset $A$ of $X$ such that $f(A) =A$. Let us first consider $A_1 = f(X)$ and recursively then ...
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Is boundedness conserved under equivalent metrics?

Let (X,$\rho$) be a general metric space where $\rho$ is a bounded metric, that is, $\exists M\in\mathbb{R}$ s.t. $\forall x,y\in X$ $\rho(x,y)<M$. Now let $\sigma$ be a metric equivalent to ...
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Example of a connected set where $\exists r>0$ such that $d(a,b) \geq r$, $\forall a \in A$, $\forall b \in B$

Our definition of separation is: If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and we cannot have that ...
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The space of strings and the Cantor sets.

We define $2^{\omega}$ as the infinite sequences of elements of the sets $\{0,1\}$ and we gives a metric $\rho (\sigma, \tau)=(\displaystyle\frac{1}{2})^{n}$ where $n$ is the length of greater finite ...
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Give a counter example to show that given two metrics are NOT equivalent.

Finding difficult to find a counterexample show that two metrics are not equivalent. Set: $C[0,1] $ of all continuous functions on the interval $[0,1]$. Metric 1: $d(x,y) = \max\limits_{t \in [0,1]} ...
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How to make sure any two points with small enough distance are inside a common open set

Let $K$ be a compact subset of a metric space, and I cover $K$ with finite open sets. How can I select $\delta>0$, such that for any $x,y\in K$, with $d(x,y) <\delta$, $x,y$ are inside an open ...
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Uniform boundedness of uniform cauchy sequence

Given $(X,d_x)$ $(Y,d_y)$ are metric spaces not neccessarily complete. Let $f_n:X \to Y$ be a sequence of maps not necessarily continuous, which is uniformly Cauchy. With for each $n\ge1$, there ...
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What do the eigenvalues/vectors of a metric describe?

Given a finite metric space $(X = \{ x_i \}_{i=1}^n,d)$, one can form the matrix $A$ of pairwise distances $a_{ij} = d(x_i, x_j)$. What does the eigenspectrum of this matrix say about the metric $d$? ...
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On the cardinality of $\mathbb R \times …\aleph_1 {times}$ and $\mathbb R \times …2^{\aleph_0} \space {times}$

I think I can prove that closure of every countable set in any metric space has cardinality at most $\mathcal c=2^{\aleph _0}$ . So if a metric space is separable i.e. has a countable dense subset $A$ ...
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A metric space of which the geodesic is not a metric

The text book in my course has an exercise about finding a metric space whose (usual) length metric is not a metric. It wants me to find a metric space $(X,d)$ satisfying $d'(x,y)=0 \ \ $for some ...
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Proving the usual distance metric in $\mathbb{R}$ is complete

If we allow the metric to be $d(x,y)=|x-y|$, we must prove that this is complete. Now, I have proven all properties of a metric space. However, I don't particularly now where to begin to prove that ...
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Confusion about notation for a metric space.

My professor wants me to prove that $(\mathbb{R},|\cdot|)$ is a complete metric. Now, I know how to do so, but I am confused as to what she is referring to by $$|\cdot|$$ Is she referring to the norm ...
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Are $\{(x,0) \in \mathbb{R^2} : x \in \mathbb{R}\}$ and $\{(x,\frac{1}{x}) : x >0\}$ separated?

Our definition of separation is: If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and we cannot have that ...
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1answer
23 views

Is it true that every 1st category subset of a 2nd category space has empty interior?

Let $X$ be a metric space. Are these conditions equivalent: Each set of the 1. category in $X$ has empty interior; $X$ is of the 2. category. It is obvious that $1 \Rightarrow 2$. Is it true that ...
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Is the following metric space is a complete metric space?

We have $X=\ell^1$, which contains sequences, which are absolutely convergent, and $d(a_n,b_n) = \sum_{k=1}^{\infty}|a_k-b_k|$. Is this metric space complete or not?
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Show the Euclidean metric and maximum metric are strongly equivalent.

I need to show that the Euclidean metric and maximum metric (or square metric??) are strongly equivalent. I have no idea how to start this proof. Any help? $d_1, d_2$ are called strongly equivalent ...
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1answer
32 views

Let $(X,d) ; (Y,e)$ be two metric spaces ; can we define a metric on $X \cup Y$ whose restriction on $X$ is $d$ and restriction on $Y$ is $e$ ?

Let $(X,d) ; (Y,e)$ be two metric spaces ; can we define a metric $\rho$ on $X \cup Y$ such that $\rho(x,y):=d(x,y) , \forall x,y \in X$ and $\rho(x,y):=e(x,y) , \forall x,y \in Y$ ?
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How to find the cardinality of the set of all isolated points of $X$?

Suppose $X$ is an uncountable complete metric space .Can the cardinality of the set of all isolated points of $X$ be countably infinite?
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If there exists an open set $U$ in $X$ such that $A = Y \bigcap U$ then $A $ is open in $Y$

Let $Y$ be subspace of a metric space $X$. Show that $A \subset Y$ is open in $Y$ if and only if there exists an open set $U$ in $X$ such that $A = Y \bigcap U$. My Try: Let $A$ be open in $Y$. Then ...
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29 views

Colorings of Topological Partitions (color-boundedness)

Let $(X,\tau)$ be a topological space. DEFINITIONS: Define a topological partition of $X$ into connected sets to be a collection of pairwise disjoint open connected sets $\{U_i\}_{i\in I}$ such that ...
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1answer
35 views

Colorings of Topological Partitions (Path adjacency)

Let $(X,\tau)$ be a topological space. DEFINITIONS: Define a topological partition of $X$ into connected sets to be a collection of pairwise disjoint open connected sets $\{U_i\}_{i\in I}$ such that ...
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2answers
38 views

$f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$

I'm trying to prove that $f:X\longrightarrow\mathbb{R}$ is continuous if, and only if, the sets $$\{x\in X:f(x)\geq\alpha\} \text{ and } \{x\in X:f(x)\leq\alpha\}$$ are closed ...
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1answer
43 views

A question about compact and complete metric spaces [duplicate]

I have a question about compact and complete metric space. These two concepts how related to each other. Is compact metric space complete? If the question is elementary I apologize you. Thank ...