Linked Questions

5 votes
0 answers
139 views

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 ...
Joel David Hamkins's user avatar
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 ...
Theo Buehler's user avatar
  • 5,703
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 ...
Dominic van der Zypen's user avatar
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 ...
Mizar's user avatar
  • 3,086
9 votes
1 answer
1k views

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 ...
Carlos's user avatar
  • 1,688
14 votes
1 answer
554 views

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 ...
Alessandro Codenotti's user avatar
6 votes
2 answers
268 views

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,\...
Dominic van der Zypen's user avatar
8 votes
1 answer
397 views

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 ...
Noam Zimhoni's user avatar
5 votes
1 answer
224 views

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\}...
D.S. Lipham's user avatar
  • 3,045
7 votes
1 answer
163 views

$2$-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 $...
Dominic van der Zypen's user avatar
5 votes
0 answers
138 views

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 ...
Taras Banakh's user avatar
  • 40.7k