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See R. Maehara, The Jordan Curve Theorem via the Brouwer Fixed Point Theorem, Amer. Math. Monthly 91, 641--643 (1984) which is availiable on Andrew Ranicki's website.

If $f$ is injective and continuous from $\mathbb{R}^n$ to $\mathbb{R}^m$ where $n>m$ then $f$ restricts to a continuous bijection from $S^{n-1}$, the unit sphere in $R^n$, to a compact subset $K$ of $ …

Yes, Sierpinski proved that every countable metric space without isolated points is homeomorphic to the rationals: http://at.yorku.ca/p/a/c/a/25.htm . An amusing consequence of Sierpinski's theorem i …

In elementary analysis one learns that every continuous function from a closed bounded interval to $\mathbb{R}$ is bounded, but this is not the case for open or unbounded intervals. A little later one …

As Andy says, each compact metric space of dimension $n$ embeds in $\mathbb{R}^{2n+1}$ (but some don't in $\mathbb{R}^{2n}$). It is the case that this extends to second countable locally compact Hausd …

For real/complex vector spaces, this is Theorem 1.21 in Rudin's Functional Analysis (2nd ed.). I believe the proof works for any complete field, but haven't checked in detail.

Every discrete space is a metric space. If we consider a measurable cardinal $\kappa$ as a discrete space, then it has an ultrafilter $\mathcal{F}$ in which the intersection of fewer than $\kappa$ ele …

It's not hard to prove Waterhouse's theorem that all profinite groups are Galois groups. Note first that each quotient of a Galois group by a normal closed subgroup is a Galois group, and as each pro …

It's certainly the case that $\mathbb{R}^2\setminus J$ is path connected. So any two points in $D\setminus J$ are joined by a path in $\mathbb{R}^2$ missing $J$. If this path isn't in $D$ it hits the …