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.
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Prove equivalent form of Baire's Category Theorem
I'm trying to prove these two statements of Baire's Category Theorem are equivalent:
Let X complete metric space. A subset of X is meagre if it can be written as the countable union of nowhere dense ...
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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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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 ...
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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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Partition a metric space into parts with small measure and diameter
Consider $\Omega = [0,1]$ with Borel $\sigma$-algebra and Lebesgure measure $\mu$. It has the property that for any $n\geq 1$, we can partition $\Omega$ into $n$ parts with small measure and diameter. ...
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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) ...
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Completeness meaning (complete basis vs complete metric space)
Today my professor started talking about the formalism of QM.
We talked about the eigenvectors of a Hermitian operator (over Hilbert space) as a "complete set". He also mentioned briefly ...
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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 ...
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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 ...
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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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Proof of existence of $\epsilon$-nets in infinite metric spaces.
The following is an excerpt from these lecture notes:
Given a metric space $(X, d)$, a subset $Y$ of $X$ is said to be an
$\epsilon$-net if
For $a, b \in Y$, we have $d(a, b) \geq \epsilon$.
For all ...
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Distances on Cartesian product [duplicate]
I was studying general topology when a question came to my mind.
Assume given n metric spaces, call $X$ their Cartesian product and define three real-valued functions from $X\times X$:
the first ...
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Proving that the set of limit points of a set is closed directly [duplicate]
I'm working on Baby Rudin chapter 2's exercises and I'm stuck on problem #6, in particular the first part where he asks to prove that the set of limit points E', of a set E, is closed.
Here's my ...
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Proving that Metric on the set of compact subsets of a metric space inherited from the metric space (Hausdorff metric) satisfies triangle inequality
On a metric space $(M,d)$ and $A \in K(M)$, the set of compact subsets of $M$, we define the function $h: M \times K(M) -> \mathbb{R}_{\geq 0}$, as $h(a,B) = \text{min} \{d(a,b): b \in B\}$, and ...
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Convergence of $\sum^{\infty }_{n=0} |a_{n}-b_{n}|$ from convergence of $\sum^{\infty }_{n=0} |a_{n}|$ and $\sum^{\infty }_{n=0} |b_{n}|$
Exercise 12.1.15 of Tao's Analysis II book asks the reader to show that the function $d:X\times X\rightarrow \mathbb{R} \cup \left\{ \infty \right\} $ defined by $d\left( a_{n},b_{n}\right) =\sum^{\...
