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This follows from Alexander duality, and works with $\mathbb{R}^2$ replaced by $\mathbb{R}^d$ for any $d$. In more detail, let us compactify $\mathbb{R}^d$ to $S^d$ and consider $F'=S^d\setminus\bigc …

A simple diagonalization argument gives a counterexample if you assume the continuum hypothesis. Let a modulus of convergence be a sequence $\delta:{\mathbb N}\to(0,\infty)$ which converges to 0, and …

I have just been teaching calculus for the first time, and I am firmly of the opinion that in many calculus courses, the mean value theorem should have essentially no role. All the applications of it …

Peter Johnstone's book Stone Spaces (p. 144) proves that for any X, maximal ideals in $C(X)$ are the same as maximal ideals in $C_b(X)$ (bounded functions), i.e. the Stone-Cech compactification $\beta …

Any map $[0,1]\to[0,1]$ is a quotient map onto its image, and the image must be either a point or a closed interval. So a partition comes from such a map iff the quotient of $[0,1]$ by the equivalenc …

Define a mean algebra to be a set $S$ with an binary operation $M$ satisfying (1), (2), and (4). We can define $M(a,b,c,d)=M(M(a,b),M(c,d))$ and this will depend only on the multiset $\{a,b,c,d\}$. …

It's not hard to construct a compact connected LOTS with this property. Let $X_0$ be any countably saturated dense linear order, and let $X$ be its bounded Dedekind completion ("bounded" meaning also …

Here is an expansion of my comment into an answer which I think is very compelling as the "correct" definition for compact Hausdorff spaces, though I agree with others who have said that for general s …