All Questions
Tagged with metric-spaces general-topology
6,085
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Prove that the usual metric and other metric induce the same topology
I am working on A course on Borel sets, by S.M. Srivastava. There is this problem I am working on that states the following:
Show that both the metrics $d_1$ and $d_2$ on $\mathbb{R^n}$ defined in 2.1....
- 329
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The function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ defined by $d\big((a,b)\big)=\lvert x-y\rvert$ is continuous [duplicate]
Let the function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ be defined by
$$
d\big( (x, y) \big) := \lvert x-y \rvert \qquad \mbox{ for all } (x, y) \in \mathbb{R}^2.
$$
Let $\mathbb{R}$ and $...
- 26.9k
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The diameter of the union of two sets in a metric space cannot exceed the sum of the diameters of the two sets and the distance between them
Let $A$ and $B$ be any two (nonempty) sets in a metric space $(X, d)$. Then how to show that
$$
d (A \cup B) \leq d(A) + d(B) + d(A, B)? \tag{0}
$$
Here we have the following definitions:
For any (...
- 26.9k
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Given a metric $d$, is continuity of $f$ absolutely essential for the composite function $f \circ d$ to be a metric (equivalent to $d$)? [closed]
Let $d$ be a metric on a nonempty set $X$, and let $f \colon [0, +\infty) \longrightarrow [0, +\infty)$ be a function such that (i) $f(0) = 0$, (ii) $f(r+s)\leq f(r) + f(s)$ for all $r, s \in [0, +\...
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Can any open set in $\mathbb{R}^d$ be countably union of closed sets
I've already know that $\{B(x,r):x\in\mathbb{Q}^d,r\in \mathbb{Q}\} $ is an countable base of $\mathbb{R}^d$. Intuitively, I wonder that can an open set $\Omega\subset \mathbb{R}^d$ be countably union ...
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Composition of asymmetric contraction mappings [closed]
Let $(M,d)$ and $(N,q)$ be metric spaces.
The operator $T:M\longrightarrow N$ is contractive in the sense that $q(T(m_1),T(m_2)) \leq c d(m_1, m_2)$ for some $c\in [0,1)$.
Similarly, the operator $J:N\...
- 162
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Intersection of interiors of sets in a partition of $\mathbb{R}^d$
Let $\mathcal{Q}=\{Q_1,...,Q_n\}$ and $\mathcal{P}=\{P_1,...,P_m\}$ be partitions of $\mathbb{R}^d$ with $n<m$. Assume all $Q_k\in\mathcal{Q}$ and $P_i\in\mathcal{P}$ are close and convex sets. ...
- 323
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Axiom of Choice in characterizing openness in subspace
Below is the typical characterization of open sets in a subspace $Y$ of a metric space $X$.
$E$ is $Y$-open iff there exists an $X$-open $S$ such that $E = S \cap Y$.
The forwards direction usually ...
- 119
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47
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Are functions from a non metrizable general topological space into $\mathbb{R}$ continous under the ring structure?
That is, would continous functions from a general topological space X be closed under field addition, multiplication and so on?
Supposing $X$ is metrizable the proof is pretty doable, and example of ...
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1
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36
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Reconciling metric and topological neighborhoods
Let $X$ be a metric space. Given a point in $x \in X$, an open neighborhood is more appropriately called an $\epsilon$-ball $N_\epsilon = \{p \in X : d(p, x) < \epsilon\}$, while a topological ...
- 119
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If a pseudonorm function $N$ is continuous in a given topology, does the pseudometric topology formed by $d(x,y)=N(x-y)$ coincide with first topology?
Define $p_n\#= p_n p_{n-1} \cdots p_1$ for $n = 0$ to be $1$, then we have a function:
$$
N : \Bbb{Z} \to \Bbb{Z}, \\
N(x) = \left |-1/2 + \sum_{d\ \mid\ p_n\#} (-1)^{\omega d} \sum_{r^2 = 1 \mod d} \...
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1
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104
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Do totally open sets exist?
In the third answer to this question, a justification is given for calling closed sets closed, since they are literally closed under the $\mathbb{N}$-ary operation of taking limits of (convergent) ...
- 3,044
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Set distances with cube neighbourhoods
This is a follow up and somewhat of a variant of the question I asked a couple of days ago (see Invariance of set distances with $\varepsilon$-neighbourhoods), and after devoting some research and ...
- 1,020
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Is Hausdorff convergence well behaved with regard to complements of sets?
Let $(X,d)$ be a compact (metric) space and $(A_n)$ a sequence of closed sets in $X$. Let $H$-$\lim_nA_n=A$ (the Hausdorff limit of $(A_n)$). Does $H$-$\lim_n(X\setminus A_n)$ exists? If yes, how is ...
- 6,349
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43
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Invariance of set distances with $\varepsilon$-neighbourhoods
I am trying to prove something involving distances between sets which I believe to be true (at least intuitively), but can't seem to get to the end. The situation is as follows. Let $\Omega$ be a non-...
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