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If you want to prove that 'complete plus densely ordered' implies connected you are almost forced to use the 'standard' proof. For the real line you could also use the bisection method: if $I$ is conv …

The answer to the secondary question is `no'. In this paper J. Terasawa constructed spaces of the form $\omega\cup\mathcal{A}$, where $\mathcal{A}$ is a maximal almost disjoint family, of arbitrary la …

By the Baire Category Theorem the full family of dense $G_\delta$-sets is a countably complete filter. So, unless I'm missing something, the set is a singleton.

And if you happen to be (able to read) Czech try Sochor's Metamatematika Teorii Mnozin

Here are a few tricks to play with. For every $A\subseteq\omega$ define $f_A$ by $f_A(0)=0$, $f_A(n+1)=f_A(n)+1$ if $n\in A$, and $f_A(n+1)=f_A(n)$ if $n\notin A$. If $A\subset^*B$ (so $B\setminus A$ …

Take a look at the table in the back of Steen and Seebach's book. You will find that Example 103 contains a completely regular space that is ccc but not separable: $\mathbb{N}^\lambda$, where $\lambda …

Why not try Weil's original paper: it's reference 12 in this paper.

You may want to have a look at this paper (PDF) by Baumgartner, Frankiewicz and Zbierski; it establishes the consistency of $\mathrm{MA}_{\sigma\text{-linked}}+\neg\mathrm{CH}$ plus ``every Boolean al …

Yes, the long ray $R$ works. If $\mathcal{U}$ is a finite open cover then $\bigcap\{R\setminus U:U\in\mathcal{U}\}=\emptyset$ and at least one of these closed sets must be bounded as in $R$ any two cl …

Rudin's Principles of Mathematical Analysis (and most every book on Mathematical Analysis) in the proof that $\lim_{x\to p}f(x)=q$ is equivalent to "$\lim_{n\to\infty}f(p_n)=q$ for every sequence $\la …

Another possibility is to use proximities - or equivalently (totally bounded) uniformities: in the metric case one defines $A$ and $B$ to be 'close' (usually denoted $A\mathrel\delta B$) if $d(A,B)=0$ …

The following is due to Alan Dow: In any model obtained by adding $\aleph_2$ many Cohen reals to a model of $\mathsf{CH}$ the statement is false. We force with $\mathbb{P}=\operatorname{Fn}(\omega_2, …

There is, in general, a on-to-one correspondence between closed subalgebras of $C^*(X)$ (the algebra of bounded continuous real-valued functions) and the compactifications of $X$. Closed in the sense …

I don't know if this is the right generalization but here is an article on Non-Archimedean probability by Benci et al (Milan J. Math. 81 (2013), no. 1, 121–151). The version of the real line used ther …

You can consult Problem 8.5.13 in Engelking's General Topology. It deals with Dieudonné complete spaces (Tychonoff spaces that have a complete uniformity). Part (d) shows that every paracompact space …