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This question arises from the excellent question posed on math.SE by Salvo Tringali, namely, Correspondence between Borel algebras and topology. Since the question was not answered there after some ti …

This question is about the space of all topologies on a fixed set X. We may order the topologies by refinement, so that τ ≤ σ just in case every τ open set is open in σ. Equivalently, we say in this c …

My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R* in place of the usual R-valued metric. That is, let us define that a topological sp …

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 …

Which are the rigid suborders of the real line? If A is any set of reals, then it can be viewed as an order structure itself under the induced order (A,<). The question is, when is this structure ri …

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 …