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Results tagged with gn.general-topology
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user 4213
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
93
votes
Accepted
Are the rationals homeomorphic to any power of the rationals?
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 …
15
votes
Accepted
Injective maps $\mathbb{R}^{n} \to \mathbb{R}^{m}$
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 $ …
10
votes
Regular borel measures on metric spaces
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 …
5
votes
Closedness of finite-dimensional subspaces
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.
4
votes
Accepted
A Jordan arc in the unit disk
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 …
4
votes
Is there a Whitney Embedding Theorem for non-smooth manifolds?
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 …
2
votes
How to understand the concept of compact space
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 …
2
votes
profinite spaces coming from profinite groups
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 …
2
votes
Pair of curves joining opposite corners of a square must intersect---proof?
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.