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Results tagged with set-theory
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user 61536
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
2
votes
1
answer
205
views
A possible characterization of regular cardinals?
For a cardinal $\kappa$ by $[\kappa]^{<\kappa}$ we denote the family of all subsets of cardinality $<\kappa$ in $\kappa$.
Question. Assume that for an infinite cardinal $\kappa$ there exists a fa …
- 41.8k
4
votes
Accepted
An almost uniform subset of the Cartesian square
At first, let us introduce some relevant definitions.
Definition. A $S\subseteq X\times Y$ is called
$\bullet$ horizontally finite in $X\times Y$ if for every $y\in Y$ the set $\{x\in X:(x,y)\in S\}$ …
- 41.8k
7
votes
1
answer
432
views
A new cardinal characteristic of the continuum?
Let $\kappa$ be the smallest cardinality of a family $\mathcal F$ of subsets of $\omega$ such that for any bijective function $f:A\to B$ between disjoint infinite subsets of $\omega$ there exists a se …
- 41.8k
9
votes
3
answers
492
views
A property of an ultrafilter
Let $\mathcal U$ be a free ultrafilter on a set $X$. For $n\in\mathbb N$ let $\mathcal F$ be a family of $n$-element subsets of $X$ such that $\bigcup\mathcal F\in\mathcal U$.
Question. Is there a …
- 41.8k
5
votes
1
answer
336
views
Is each cosmic space cometrizable?
A regular topological space $X$ is called
$\bullet$ cosmic if $X$ is a continuous image of a separable metrizable space;
$\bullet$ cometrizable if $X$ admits a weaker metrizable topology such that e …
- 41.8k
2
votes
0
answers
116
views
Cardinal characteristics of the ideal of $\sigma$-continuity of the Pawlikowski function
A function $f:X\to Y$ between topological spaces is called $\sigma$-continuous if there exists a countable cover $\mathcal C$ of $X$ such that for every $C\in\mathcal C$ the restriction $f{\restrictio …
- 41.8k
7
votes
1
answer
365
views
The diamond principle for functors
Let $F:\mathbf{Comp}\to\mathbf{Set}$ be a continuous functor from the category of compact Hausdorff spaces to the category of sets such that $|Fn|\le\mathfrak c$ for any finite ordinal $n$. The contin …
- 41.8k
11
votes
1
answer
416
views
A monotone countably cofinal function from $\omega^\omega$ to $\omega^{\omega_1}$
For a set $X$ we endow the set $\omega^X$ of all functions from $X$ to $\omega$ with the natural partial order $\le$ defined by $f\le g$ iff $f(x)\le g(x)$ for all $x\in X$.
A function $f:\omega^\ome …
- 41.8k
4
votes
Possible cardinality and weight of an ordered field
Browsing through MathOverflow I have found some answers to related questions, which shed light on my question too. It turned that my question has been considered in the literature. The most appropriat …
- 41.8k
3
votes
1
answer
155
views
An infinite subset in $3^\omega$ with "large" projections?
Problem. Is there an infinite set $I\subset 3^\omega$ such that for any infinite subset $J\subset I$ there exists $n\in\omega$ such that $\{x(n):x\in J\}=3$?
Here $3^\omega$ is the set of functions f …
- 41.8k
1
vote
Accepted
A monotone countably unbounded function from $\omega^\omega$ to $\omega^{\omega_1}$
The answer to this problem is negative and follows from
Theorem. For any monotone function $\mu:\omega^\omega\to\omega^{\omega_1}$ there exists a countable infinite set $A\subset\omega_1$ such that f …
- 41.8k
7
votes
Accepted
Cardinality of a set of countable connected Hausdorff spaces
It seems that the number of such topologies is $2^{\mathfrak c}$. Such (huge) number of connected Hausdorff topologies can be constructed by a suitable modification of the Bing's construction of a con …
- 41.8k
12
votes
Accepted
Surjective group homomorphism from $\text{Sym}(X)$ onto $\mathbb{Z}$
The answer here is negative. In fact, any non-trivial quotient group of the symmetric group $\mathrm{Sym}(X)$ contains a copy of $\mathrm{Sym}(X)$. Indeed, by the Baer-Schreier-Ulam Theorem,
any norm …
- 41.8k
6
votes
If $(X,\tau)$ has more than $1$ point and is $T_2$ and connected, do we have $|X| =|\tau|$?
The Golomb space $(\mathbb N,\tau)$ (a "universal" counterexample to many questions), also gives a counterexample with $|\mathbb N|=\aleph_0$ and $|\tau|=\mathfrak c$.
I recall that the Golomb space …
- 41.8k
6
votes
Accepted
Connected spaces where every dense set is large
Yes. Take any countable connected Hausdorff space $C$, fix any point $c\in C$, and consider the quotient space $X=C\times \kappa/ \{c\}\times \kappa$. Here the cardinal $\kappa$ is endowed with the di …
- 41.8k