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The answer is yes. My original argument made use of the continuum hypothesis, or actually just the assumption that $2^\omega<2^{\omega_1}$), but this assumption has now been omitted by the argument o …

The answer is Yes. Theorem. The following are equivalent for any Hausdorff space $X$. $X$ is compact. $X^\kappa$ is Lindelöf for any cardinal $\kappa$. $X^{\omega_1}$ is Lindelöf. Proof. The forw …

Yes, there is such a space. Let $X=2^{\omega_1}$ be the space of binary sequences of length $\omega_1$, in the order topology generated by the lexical order. So $X$ consists of the branches through th …

Todd has already answered the question, but let me give an alternative argument. Theorem. Every compact Hausdorff space of size less than the continuum is totally disconnected. Proof. Suppose $a\neq …

For any irrational number $x$, let $f(x)$ be the real number arising from the integer part of $x$, together with every other digit of the rest of the expansion of $x$. This is surjective, since one …

There is some fascinating work in the subject of cardinal characteristics of the continuum in set theory that directly relates to the concept of growth rates of functions. I believe that it is the ide …

The answer is that the FPP is not absolute, and indeed, even the unit interval loses the FPP in a forcing extension. The unit interval famously has the FPP, but I claim that in any forcing extension h …

If you allow the functions to be constant on some intervals, then there are some easy examples, and Ricky has provided one. But if you rule that out, then there can be no examples, even with countab …

This is a great question! The disjoint union of two circles is $1$-homogeneous, but not $2$-homogeneous. It is $1$-homogenous, since you can swap any two points and extend this to a homeomorphism (ba …

Theorem. There is no chain of nowhere dense subsets of $\mathbb{R}$ whose union contains an interval. Proof. Suppose there was such a chain $\{\ B_i \mid i\in I\ \}$, where $\langle I,\lt\rangle$ is …

Let $D_0,D_1,\ldots$ enumerate a sequence of disjoint intervals in the unit interval with $\bigcup_n D_n$ open dense and having measure less than $1$. For example, place a very tiny interval around ea …

Your set $\partial\mathbb{N}$ is also intensely studied in set theory and known as P(ω)/Fin. What you have done is mod out by the ideal of finite sets. People study more general properties P(X)/I, tak …

Nice question! I claim that this property does not necessarily imply CH. As Todd guessed in his comment, the answer is related to certain cardinal characteristics of the continuum. Specifically, let …

Every analytic set ($\Sigma^1_1$ set) of reals is the projection of a Borel subset of $\mathbb{R}\times\mathbb{R}$, and the projection map $p(x,y)\mapsto x$ is an open map. So the standard examples of …

The set of all topologies on a given set $X$ admits a lattice structure under the refinement relation $\tau\leq\sigma$, whereby every $\tau$ open set is open with respect to $\sigma$, meaning that $\s …