Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space. For questions about Riemannian metrics use the tag (riemannian-geometry) instead.

A function $d: M\times M\to \mathbb R$ is called a metric if for all $x,y,z \in M$ we have

  1. $d(x,y)=0\iff x=y$
  2. $d(x,y)\geq 0$
  3. $d(x,y)=d(y,x)$
  4. $d(x,y)+d(y,z)\geq d(x,z)$.

It is a generalisation of "distance". A metric space is now defined as an ordered pair $(M,d)$, where $M$ is a set and $d:M\times M\to R$ is a metric.

An $\varepsilon$-neighbourhood of $x$ is defined as the set $$B_\epsilon(x):=\{y\in M\mid d(x,y)<\varepsilon\}.$$ $B_\varepsilon(x)$ is commonly also known as the open ball of radius $\varepsilon$ around $x$. All open balls form a base for a topology on $M$. Although all metric spaces are topological spaces, the converse is generally not true.

Some different types of metric space include

  1. Complete metric spaces (every Cauchy sequence converges)

  2. Bounded metric spaces (every metric is bounded by a finite value)

  3. Compact metric spaces (every sequence has a convergent subsequence)

  4. Locally compact metric spaces (every point has a compact neighbourhood)

  5. Separable metric spaces (it possesses a countable dense subset).

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Metric spaces and fix point

I saw this problem in my course of Intr. to the topology: Let $(X,d)$ be a compact metric space and $$f :(X,d) \rightarrow (X,d)$$ a continuous function such that: $\quad d(f(x);f(y)) < d(x;y)$, if $x\neq y$. Prove that $f$ has an unique fix…
Ysaac
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connectedness vs. path connectedness

Is there a general rule of what kind of sets it is easier to prove connectedness using path connectedness or regular connectedness? I understand that path connected $\implies$ connected, but are there situations where it's easier to prove using…
Emir
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Path-connectedness of continuous functions

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 outcome. I simply tried the straight line path $p(t) =…
surfer1311
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A strange criterion for compactness

Is it true that if every continuous real-valued function on a metric space is bounded, then that metric space is compact
user168826
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Assume that the only dense subset is $X$ itself. What can you say something about the topology, that is, the family of open sets?

Let $(X, d)$ be a metric space. Assume that the only dense subset is $X$ itself. What can you say something about the topology, that is, the family of open sets? My Try: Since the only dense subset of $X$ is $X$ itself, so every subset of $X$ is a…
User8976
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prove $\mathbb{N}$ is complete w.r.t. $d_2$

Prove $(\mathbb{N},d_2)$ is a complete metric space. Attempt: So I need to show that every Cauchy sequence in this metric space converges. Presumably all of these convergent Cauchy sequences would be eventually constant -- otherwise they wouldn't…
Emir
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Proving the Cantor Set contains no segment

I know that the Cantor Set contains no segment of the form $$\left(\frac{3k+1}{3^m}, \frac{3k+2}{3^m}\right)$$ for any integers $k$ and $m$. If we can prove that every real segment contains a segment of that form, then certainly the Cantor Set…
jamaicanworm
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Is the sum of two complete metrics complete?

Let $X$ a space with two complete metrics $d_1$, $d_2$: Is $d=d_1+d_2$ complete?
JPaucar
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Definition of equicontinuity of a mapping between metric spaces

From Wiki Let $X$ and $Y$ be two metric spaces, and $F$ a family of functions from $X$ to $Y$. The family $F$ is equicontinuous at a point $x_0 ∈ X$ if for every $ε > 0$, there exists a $δ > 0$ such that $d(f(x_0), f(x)) < ε$ for all $f ∈ F$…
Tim
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$A$ is open $\iff$ $A\cap\partial A=\emptyset$

Show $A$ is open $\iff$ $A\cap\partial A=\emptyset$. Attempt: ($\rightarrow)$ $A$ open $\implies A\cap\partial A= \emptyset$. $x\in A$ open $\implies\exists\epsilon>0:B_{\epsilon}(x)\subseteq A$ $\implies$ $B_{\epsilon}(x)\cap…
Emir
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To find a counterexample in metric space.

Suppose $X$ is a metric space, $z \in X$ and $(x_n)$ is a sequence in $X$. Show that if $X$ has a subsequence that converges to $z$, then dist$(z ,$ {$x_n :n ∈ N$}) $= 0$, and show also that the converse need not be true. I have proved the first…
User8976
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If $(a_n)$ is Cauchy it has a subsequence $(a_{n_i})$ such that $d(a_{n_{i+1}},a_{n_i})<2^{-i}$ for all $i$.

Let $(X,d)$ be a metric space and $(a_n)$ a Cauchy sequence in $X$. How to show that there exists $n_1
Steph
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Projection map is open

Let $X=X_1$ x $X_2$ where $(X_1,d_1)$ and $(X_2,d_2)$ are metric spaces. Equip $X$ with a product metric $d$. Define a map $\Pi_1:X \to X_1$ by $\Pi_1(x_1,x_2) = x_1$. Let $U \subset X$ be open and let $x_0 \in U_1 = \Pi_1(U)$. As $x_0 \in U_1…
user60327
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Metric Spaces, Continuity and Preference Relations

Let X be a metric space and $\succeq$ be a preference relation on X. The preference relation is continuous if the sets $\succeq (y) =\{x: x \succeq y\}$ and $\preceq (y) = \{x : x \preceq y\}$ are closed for every $y$. Assume that $B \subseteq X$ is…
user171891
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showing completeness of a metric space

$X=\mathbb{R}_{>0}$, $d(a_1,a_2)=|\ln(a_1)-\ln(a_2)|$. I have already proven that $(X,d)$ is a metric space, but I have some problems showing the completeness. Let $(a_n)_{n\in\mathbb{N}}$ a Cauchy series in $X$, which means $|\ln(a_n)-\ln(a_m)|…
Val
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