# Tag Info

Accepted

• 17.3k

### Diagonalizing against $\omega_1$-sequences of functions mod finite

This isn't an answer, as you're working in ZFC. But it seems worth noting. Assume ZFC + AD$^{L(\mathbb{R})}$. Then $L(\mathbb{R})$ satisfies ZF + AD + DC + "the statement is false". Proof: ...
• 8,627
Accepted

### $\operatorname{cof}(\mathcal{L}) = \aleph_1$ and $2^{\aleph_0} = \aleph_3$

Yes, this is consistent. Probably the simplest way to get a model of this theory is to begin with a model of GCH, and then force with a countable support product of $\aleph_3$ copies of the Sacks ...
• 17.3k
Accepted

### Is there a condensation from $\aleph_1^{\aleph_0}$ onto a metrizable compact space?

If $|D| < \aleph_\omega$, then there is a condensation from $D^\omega$ onto $\omega^\omega$ (the Baire space) if and only if there is a partition of $\omega^\omega$ into exactly $|D|$ Borel sets. ...
• 17.3k
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• 13.9k

### Is there a condensation from $\aleph_1^{\aleph_0}$ onto a metrizable compact space?

The answer is affirmative and can be derived from Theorem (Banakh, Plichko). The Hilbert space $\ell_2(\aleph_1)$ condenses onto the Hilbert cube. By the way, this theorem is related to Problem 1 ...
• 40.4k
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• 44.4k
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### Pseudo-intersections, splitting families, and ultrafilters

The answer is no -- it is consistent that every $U \in \omega^*$ has $\tau(U) < \mathfrak{s}$. I had an idea for proving this earlier today, using the Mathias model. I couldn't quite make things ...
• 17.3k

• 17.3k
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• 2,560
Accepted

• 17.3k
Accepted

### Comparing bornologies for cardinal characteristics via Borel maps

A bornomorphic map $i\colon\mathcal{N}\to\mathcal{N}$ for these bornologies cannot be Borel. First, I shall use $f<^\infty g$ to mean that $(\forall m\in\mathbb{N})(\exists n>m)f(n)<g(n)$, ...
• 1,251
No, these cardinals are not provably equal. This follows from a result of El'kin 1: Let $X$ be a set and let $x \in X$. If $\tau$ is a maximal topology on $X$, then \$\{U \setminus \{x\} \,:\, x \...