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Actually real open intervals with rational left end-point and irrational right end-point are a base with that property.

Any countably compact space $X$ has this property. The image of a continuous real valued function of $X$ is a countably compact subset of $\mathbb{R}$, hence compact because it is Lindelöf, so it has …

It is worth remarking that the analogous characterization of σ-algebras also holds in the case of countable underlying sets: Any σ-algebra $\mathcal{A}$ on a countable set $S$ is atomic. That …

I just invented this "uglification of $\mathbb{N}$" for you. Shouldn't it work, at least, the name will be justified. Let $A:=\mathbb{N}$ and let $B$ be the quotient set of $\mathcal{P}(\mathbb{N})$ …

The following gives a partial answer: no such unbounded connected set may exist with the further assumption that it is closed. Actually, the argument generalizes for any locally compact metric space. …

The only meaning of the statement I can see is, the (quite obvious) fact that a 90 degrees rotation of the graph of a continuous, increasing function $f$ gives the graph of a function, plus the jumps. …

No, take $S=[-2,2]\times[-2,2]$ and $F$ the closure of the graph of $[-2,2]\setminus\{0\}\ni x\mapsto \sin(1/x)$. $$*$$ However, it is true that there is a connected component of $F$ that meets both t …

What you are describing is exactly the ball measure of non-compactness of the closed bounded set $B$: $$\alpha(B):=\inf \{r>0 \, : \exists F\subset B, \mathrm {\, F\, finite\, ,s.t.} B\subset \cup_{x\ …

The property you mention does not imply the existence of a convergent subnet: take e.g. the sequence $x_n:=\sqrt n$ on $\mathbb{R}$, as a metric space with the truncated standard distance, $d(x,y):=\m …

Note that the case of real-valued functions is easy. A function $f:A\to\mathbb{R}$ has a uniformly continuous extension to any metric space $X\supset A$ iff it has a sub-linear modulus of continuity …

In fact it is not true in this generality: the simple counterexample, inspired by a famous Aesop's fable, is: $X=\mathbb{R}$, $f(x)=x$, and $g(x)=\max(0, 2x-1)$ for $x\in I:=[0,1]$. The Fréchet distan …

No: if $X$ is non compact, it is a proper and dense subset, thus not closed, in its Stone–Čech compactification. [edit] This is ok e.g. if X is $T_{3.5}$, as Bruno observes, otherwise $X\to\beta X$ …

Let $B_r$ denote the closed ball of radius $r$ centered at $0$ in $\mathbb{R}^d$. Claim: for any $r>0$ and $\tau <T$ there is a countable subfamily of $D$ that covers $[0,\tau] \times B_r$. To pro …

Certainly, some of these paths may connect a couple of points $a$ and $b$ on the same fiber of $f$, $f(a)=f(b)=c\in\mathbb{S}^{n-1}$, but at one of them $f$ preserves the orientation, and exchanges …

Yes (I assume $A$ has the induced topology). The point $V:=(0,1)$, common endpoint of all segments $S_r:=\{(rt,1-t): t\in[0,1]\}$, is either fixed by the continuous function $f:A\to A$, or it is mappe …