general topology - When are compact metric spaces connected ...
Jan 9, 2023 · Prove that for a compact metric space $X,$ [$X$ cannot be written as $X=A\\cup B$ for nonempty subsets $A$ and $B$ st $d(A,B)>0$] iff $X$ is connected The question ...
general topology - In which cases connectedness and path …
Apr 16, 2021 · Every locally compact, connected metric space is separable, so define a generalized Peano continuum to be a locally compact, locally connected, connected metric …
compact connected space is uncountable?
Show that a compact connected metric space with more than one point is uncountable. Ive seen some proofs that it works without discussing compactness. But I am after a proof that is about …
Can an "almost injective'' function exist between compact …
Oct 12, 2020 · @Teri take a sequence of points x_k in your set N that converges to a point x in the northern hemisphere, but not in N. Take the corresponding points y_k in the southern …
Connectedness of balls in a compact, connected metric space
Jul 22, 2019 · Let $ (X,d)$ be a compact, connected metric space. For every $\epsilon>0$ define an equivalence relation on $X$ by $x\sim_ {\epsilon}y$ if and only if there exists a ...
Metric space - Encyclopedia of Mathematics
Mar 25, 2023 · Moreover, in each metric space there is a base such that each point of the space belongs to only countably many of its elements — a point-countable base, but this property is …
Clearly every explosion point is also a dispersion point, but the converse is not true. Indeed, in 1921, Knaster and Kuratowski first gave an example of a space having a dis-persion point, …
Metric space - Encyclopedia of Mathematics
Moreover, in each metric space there is a base such that each point of the space belongs to only countably many of its elements — a point-countable base, but this property is weaker than …
In this work, we completely characterize planarity for compact, locally connected metric spaces. Such a space has only finitely many components. If none of them is a sphere, then the whole …
Solving this problem involves a restudy and extension to general spaces of many of the results about cut points, conjugacy of point pairs, structure of locally connected spaces relative to its …
A locally compact, connected metric space is $\sigma$-compact
Dec 30, 2016 · A locally compact, connected metric space is $\sigma$-compact Ask Question Asked 8 years, 7 months ago Modified 8 years, 7 months ago
Connected metric spaces with at least 2 points are uncountable.
That's a problem I proved (quite a while back) in tiny Rudin. However, I don't really get it. The other questions were actually useful results - I don't think I've ever come near using this result.
Prove if $X$ is a compact metric space, then $X$ is separable.
And it is because if you want a point in the set within $\varepsilon$ of a given $x_0$, the center of one of the balls you made at the $n$th step will work, provided that $n>1/\varepsilon$.
general topology - Homeomorphism preserving distance
May 22, 2012 · A point $x$ is called a non-cut point if $M\setminus\ {x\}$ is connected. Every nontrivial connected compact metric space has at least two non-cut points (see Analytic …