# 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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### 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 $\rho$....
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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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### 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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### 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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### 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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### 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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### 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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### 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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### Pointwise convergence imply uniform convergence

I am trying to find a condition under which a sequence of continuous functions on a metric space (or more generally in a topological space) which point wise converge to some function f should imply ...
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### Understanding proof that if $c_1 d_1(x,y) \leq d_2 (x,y) \leq c_2 d_1 (x,y)$ then $d_1$ and $d_2$ are topologically equivalent metrics

Theorem. If there are strictly positive constants $c_1$ and $c_2$ such that $$c_1 d_1(x,y) \leq d_2 (x,y) \leq c_2 d_1 (x,y)$$ for all $x,y \in X$, then $d_1$ and $d_2$ are topologically ...
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### Problem in standard proof of continuity when pre-image is open?

I have seen several proofs of the fact that a function $f$ from a metric space $X$ to a metric space $Y$ is continuous if every open set on $Y$ has an open inverse image on $X$. When proving the ...
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### Two norms $||.||_1$ and $||.||_2$ on a vector space $V$ are equivalent.

Two norms $||.||_1$ and $||.||_2$ on a vector space $V$ are equivalent iff there exist positive constants $C_1,C_2$ such that $$C_1||.||_1 \leq ||.||_2 \leq C_2||.||_1$$ for all $x \in V$. I have ...
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### If $A$ , $B$ are dense in the metric space $X$ then,…

Let $X$ is a metric space and $A$ and $B$ are two dense subset in $X$. Which is correct? if $A$ is open, $A‎ \cap‎‎B$ is dense in $X$ if $A$ is closed in $X$, $A‎ \cap‎‎B=\emptyset$ $(A-B)\cup(B-A)$ ...
I want to prove that the metric space $C[0,1]$ with the metric $d(f,g) = sup_{x \in [0,1]} |f(x) - g(x)|$ is path-connected. I think I've done most of the proof, but I am not too sure about the ...
### Proving that the metric space $((0,\infty),d)$ is complete, with $d(x,y)=|\ln x-\ln y|$ [duplicate]
Let $X$ denote $(0,\infty)\subseteq \mathbb{R}$, and let $d:X\times X\to \mathbb{R}$ be defined as $d(x,y)=|\ln x- \ln y|$. Show that $(X,d)$ is a complete metric space. I am taking for granted here ...