Intereting Posts

What went wrong?
Why $\mathbf{0}$ vector has dimension zero?
Book suggestion for probability theory
Determine the closure of the set $K=\{\frac{1}{n}\mid n\in\mathbb N\}$ under each of topologies
Proof on minimal spanning tree
A minimization problem in function fitting setup
$H_0(X,A) = 0 \iff A$ meets each path-component of $X$.
Solving the recurrence $T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{2}\log(n)$
Maximal ideals in polynomial rings with real and complex coefficients
If $\{a_1,\dots,a_{\phi(n)}\}$ is a reduced residue system, what is $a_1\cdots a_{\phi(n)}$ congruent to?
What does the $\phi^{-1}$ mean in this formula: $\rho_y(y)=\rho_x(\phi^{-1}(y))\left|\frac{{d}\phi^{-1}}{{d}y}\right|$?
Lower Semicontinuity Concepts
Geometric multiplicity of an eigenvalue
Are there infinitely many primes of the form $n^2 – d$, for any $d$ not a square?
Can you give me an example of topological group which is not a Lie group.

Similar to the Egoroff theorem,we can get the following theorem:

\textbf{Theorem:} Let $X$ be a locally compact Hausdorff space(non-empty),and $\{f_n\}$ be a pointwise bounded sequence of continuous functions defined on $X$,then $\{f_n\}$ are bounded uniformly on an open subset of $X$.

From this,I want to judge whether the theorem is hold for pointwise convergent sequence:

Let $X$ be a locally compact Hausdorff space(non-empty),and $\{f_n\}$ be a pointwise convergent sequence of continuous functions defined on $X$,then $\{f_n\}$ are convergent uniformly on an open subset of $X$.

Since a bounded sequence can contain a convergent subsequence,I guess the above theorem is also true,but I don’t know where to start.

- Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice?
- Generating the Sorgenfrey topology by mappings into $\{0,1\}$, and on continuous images of the Sorgenfrey line
- Axiom of choice and compactness.
- Showing the metric $\rho=\frac{d}{d+1} $ induces the same toplogy as $d$
- Intuition behind topological spaces
- induced map homology example

- cauchy sequence on $\mathbb{R}$
- Rudin assumes $(x^a)^b=x^{ab}$(for real $a$ and $b$) without proof?
- How to show $\gamma$ aka Euler's constant is convergent?
- Arcwise connected part of $\mathbb R^2$
- Why is that the extended real line $\mathbb{\overline R}$ do not enjoy widespread use as $\mathbb{R}$?
- Bounded sequence with divergent Cesaro means
- For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$
- How to prove $\lim_{n \to \infty} (1+1/n)^n = e$?
- Is every compact subset of $\Bbb{R}$ the support of some Borel measure?
- A continuously differentiable function with vanishing determinant is non-injective?

Here is another example (perhaps in fact the same as David’s). It is a bit too long for a comment.

Enumerate all non trivial closed sub-intervals of $[0,1]$ with rational endpoints as a sequence $(I_k)_{k\in\mathbb N}$. For each $k$, choose your favourite sequence of continuous functions converging pointwise to $0$ but not uniformly on $I_k$. More precisely, choose a sequence of continuous functions $(f_{n,k})_{n\in\mathbb N}$ on $[0,1]$ with $f_{n,k}\equiv 0$ outside $I_k$ and $0\leq f_{n,k}\leq 1$, such that $f_{n,k}\to 0$ pointwise as $n\to\infty$ but $\sup_{x\in I_k} f_{n,k}(x)=1$ for any $n,k$.

Then define $f_n(x)=\sum_{k=1}^\infty 2^{-k}f_{n,k}(x)$. The $f_n$ are continuous (uniform convergence) with $0\leq f_n\leq 1$. It is not hard to check that $f_n\to 0$ pointwise (you have to interchange a limit and a $\Sigma$). On the other hand, for any $k, n$ you have $f_n(x)\geq 2^{-k} f_{k,n}(x)$; so $\sup_{x\in I_k} f_n(x)\geq 2^{-k}$ for each $k$ and all $n\in\mathbb N$, and hence no subsequence of $(f_n)$ can converge uniformly to $0$ on any $I_k$. Since any open subset of $[0,1]$ contains an $I_k$, it follows that no subsequence of $(f_n)$ converges uniformly on any open set.

- Biggest powers NOT containing all digits.
- Simplicity and isolation of the first eigenvalue associated with some differential operators
- $GL(n,K)$ is open in $M(n,K)$
- Why is an orthogonal matrix called orthogonal?
- Is every countable dense subset of $\mathbb R$ ambiently homeomorphic to $\mathbb Q$
- Doubt Concerning the Definition of a Locally Convex Space Structure through Seminorms?
- Morphism between projective schemes induced by injection of graded rings
- When does a null integral implies that a form is exact?
- $\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$
- $R\subseteq S$ integral extension and $S$ Noetherian implies $R$ Noetherian?
- Non-revealing maximum
- elementary set theory (cartesian product and symmetric difference proof)
- General Proof for the triangle inequality
- Linear operator on normed space
- Irreducibility issue