Linked Questions
5
votes
0
answers
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How big is the smallest nontrivial partition of the unit interval into closed disjoint closed sets? [duplicate]
Consider how we might partition the unit interval in the reals into
disjoint closed sets
$$[0,1]=\bigsqcup_i C_i.$$
Of course, we could partition the unit interval into singletons, which would make ...
34
votes
4
answers
8k
views
Why are the integers with the cofinite topology not path-connected?
An apparently elementary question that bugs me for quite some time:
(1) Why are the integers with the cofinite topology not path-connected?
Recall that the open sets in the cofinite topology on a ...
19
votes
3
answers
1k
views
"Anti" fixed point property
Let $(X,\tau)$ be a topological space. If $f:X\to X$ is continuous, we say $x\in X$ is a fixed point if $f(x) = x$.
The space $(X,\tau)$ is said to have the anti fixed point property (AFPP) if the ...
11
votes
1
answer
486
views
Is $[0,1]$ a disjoint union of $\aleph_1$ compact subsets with empty interior?
Is $[0,1]$ a disjoint union of $\aleph_1$ compact subsets with empty interior?
The answer is obviously yes assuming the continuum hypothesis. Also, by Baire's lemma, the answer is negative if one ...
9
votes
1
answer
1k
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Uncountable disjoint closed coverings of $[0,1]$
It is well known that the unit interval $[0,1]$ cannot be decomposed as a countable union of pairwise disjoint closed (nonempty) subsets. See for instance this math.stackexchange question. The proof ...
14
votes
1
answer
554
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How “disconnected” can a continuum be?
A continuum is a compact connected metrizable topological space.
Given a cardinal $\kappa$, a topological space $X$ is called $\kappa$-connected if it is not possible to write $X$ as the disjoint ...
6
votes
2
answers
268
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Infinite "almost rigid" homogeneous $T_2$-space
A topological space $(X,\tau)$ is said to be homogeneous if for all $x,y$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$.
Is there an infinite homogeneous Hausdorff space $(X,\...
8
votes
1
answer
397
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Is a cofinite topology for a set with cardinality between $\aleph_{0}$ and $2^{\aleph_{0}}$ path-connected?
It is easy to show that $\mathbb{N}$ with the cofinite topology is not path connected and that any set with cardinality $\geq 2^{\aleph_0}$ equipped with the cofinite topology is in fact path ...
5
votes
1
answer
224
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How many disjoint compact sets are needed to form a connected compactum?
Let's assume all spaces are metrizable. For each connected compact space $X$ let $\mathscr K(X)$ be the set of all partitions of $X$ into non-empty compact sets, excluding the trivial partition $\{X\}...
7
votes
1
answer
163
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$2$-determined Hausdorff spaces
Is there an infinite Hausdorff space $(X,\tau)$ with the following property?
If $x\neq y \in X$ and $f:\{x,y\}\to X$ is a map, then there is exactly one continuous function $f': X\to X$ such that $...
5
votes
0
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Disjoint covering number of an ideal
Let $\mathcal I$ be a $\sigma$-ideal with Borel base on an uncountable Polish space $X=\bigcup\mathcal I$.
Let $\mathrm{cov}(\mathcal I)$ (resp. $\mathrm{cov}_\sqcup(\mathcal I)$) be the smallest ...