Intereting Posts

Puiseux Series?
A proof of $\int_{0}^{1}\left( \frac{\ln t}{1-t}\right)^2\,\mathrm{d}t=\frac{\pi^2}{3}$
Most general $A \subseteq \mathbb R$ to define derivative of $f: A \to \mathbb R$?
Evaluate a finite sum with four factorials
$g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$
on the adjointness of the global section functor and the Spec functor
Generating orthogonal axes on a spline
Evaluation of $\int_0^1 \frac{\log^2(1+x)}{x} \ dx$
At what points is the following function continuous?
What is the best way to calculate log without a calculator?
Decidability of tiling of $\mathbb{R}^n$
There are $12$ stations between A and B, in how many ways you can select 4 stations for a halt in such a way that no two stations are consecutive
Eigenvalue problems for matrices over finite fields
Proof of the formula for Euler's totient function
Proof there is a 1-1 correspondence between an uncountable set and itself minus a countable part of it

Suppose $E$ is a simply-connected open subset of $\mathbb R^n$. Must there be a sequence of compact subsets $K_n$ such that $E = \bigcup_{n=1}^\infty K_n$, $K_n \subseteq K_{n+1}$ for all $n$, and each $K_n$ is simply connected? This is trivially true if $n=1$, and is true if $n=2$ by the Riemann mapping theorem, but topology can become counter-intuitive in higher dimensions.

- Prove existence of disjoint open sets containing disjoint closed sets in a topology induced by a metric.
- Topological counterexample: compact, Hausdorff, separable space which is not first-countable
- Why does Totally bounded need Complete in order to imply Compact?
- universal property in quotient topology
- Show for all continuous $f: (\mathbb{N}, \tau_{c}) \to (\mathbb{R}, \tau_{ST})$, $f$ is a constant function
- Continuity of argmax (moving domain and function), under uniqueness hypothesis
- Continuity based on restricted continuity of two subsets
- $X \times Y$ homeomorphic to $Z \times Y$ implies $X$ is homeomorphic to $Z$?
- Where is the Hausdorff condition used?
- Metric is continuous function

This is false in dimension 3, Whitehead manifold is an example.

One way to see it is to observe that simply connected compact 3-manifolds with connected nonempty boundary are balls, while Whitehead manifolds cannot be exhausted by balls.

Here is a more general theorem:

**Theorem.** Suppose that $M$ is a simply-connected 1-ended connected 3-dimensional manifold which is not *simply-connected at infinity* (e.g., Whitehead manifold). Then $M$ does not admit an exhaustion by simply-connected compact submanifolds with boundary.

Proof. The assumption that $M$ is not simply-connected at infinity means that there exists a compact subset $K\subset M$, such that for every compact subset $K’\subset M$ containing $K$, there exists a loop $\gamma\subset M – K’$ which does not contract in $M-K$. Suppose that such $M$ admits an exhaustion by simply-connected compact submanifolds with boundary $M_i$. Then the boundary of each $M_i$ is a disjoint union of 2-spheres. It follows that for each component $C_i$ of $M-M_i$, the boundary of $C_i$ is a 2-sphere. Therefore, $\pi_1(C_i)\to \pi_1(M)$ is injective. But this contradicts the property that $M$ is not simply-connected at infinity. qed

**Note.** In the solution I assumed that you are only asking about exhaustions by compact submanifolds with boundary. (The proof also works if you exhaust by simplicial subcomplexes.) Exhaustions by arbitrary compact subsets would be much more difficult to analyze and I am not sure how to solve the problem in this case.

- Show quadratic equation has two distinct real roots.
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- How to prove that $(x-1)^2$ is a factor of $x^4 – ax^2 + (2a-4)x + (3-a)$ for $a\in\mathbb R$?
- Expanding a potential function via the generating function for Legendre polynomials
- functions $f=g$ $\lambda$-a.e. for continuous real-valued functions are then $f=g$ everywhere
- Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$?
- Why can't Fubini's/Tonelli's theorem for non-negative functions extend to general functions?
- Cross product in complex vector spaces
- What is a primitive point on an elliptic curve?
- Does the sum $\sum_{n=1}^{\infty}\frac{\tan n}{n^2}$ converge?
- Splitting Field over a Field
- About stationary and wide-sense stationary processes
- A finite group which has a unique subgroup of order $d$ for each $d\mid n$.
- Span of Permutation Matrices
- What are some common pitfalls when squaring both sides of an equation?