Intereting Posts

What is the Modulus of a Matrix?
A coin is ﬂipped 8 times: number of various outcomes
Do there exist an infinite number of 'rational' points in the equilateral triangle $ABC$?
When linear combinations of independent random variables are still independent?
Dot Product Intuition
Mutual Independence Definition Clarification
Sequence $a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n$.
How to prove $8^{2^n} – 5^{2^n}$ is divisible by $13$ where $n\in\mathbb{N}-\{0\}$ with induction?
Find the general solution of the differential equation $\left(3y^2+x^2+x+2y+1\right)\cdot y'+2xy+y=0$
Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $$
Roots of $ x^3-3x+1$
Finding a singluar cardinality
Conformal map from unit disk to strip
Rational quartic curve in $\mathbb P^3$
Easy Proof Adjoint(Compact)=Compact

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.

- Looking For a Neat Proof of the Fact that the Grassmannian Manifold is Hausdorff
- Hairy Points in Infinite Graphs (and Peano Continua)
- Why does Seifert-Van Kampen not hold with $n$-th homotopy groups?
- Choosing a text for a First Course in Topology
- partial converse of existence of covering spaces
- The automorphism group of the real line with standard topology

- Quotient space of closed unit ball and the unit 2-sphere $S^2$
- How to prove boundary of a subset is closed in $X$?
- Topological properties of symmetric positive definite matrices
- Bound on first derivative $\max \left(\frac{|f'(x)|^2}{f(x)} \right) \le 2 \max |f''(x)|$
- Continuous mapping on a compact metric space is uniformly continuous
- Proving separability of the countable product of separable spaces using density.
- Proving $\lim_{n\to\infty} a^{\frac{1}{n}}=1$ by definition of limit
- Continuity of a function at an isolated point
- A continuous map that fixes the boundary of a domain pointwise is surjective
- Construct a function that takes any value even number of times.

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.

- A characterisation of quadratic extensions contained in cyclic extensions of degree 4
- Stuck on proving uniform convergence
- Expectation of norm of a random variable
- How do we take second order of total differential?
- Words built from $\{0,1,2\}$ with restrictions which are not so easy to accomodate.
- Trust region sub-problem with Jacobi Condition
- How to prove $\vdash\neg P\to (P\to Q)$?
- Proofs in Limits
- A problem on positive semi-definite quadratic forms/matrices
- Why must the decimal representation of a rational number in any base always either terminate or repeat?
- Is the set of all numbers which divide a specific function of their prime factors, infinite?
- Why is 'catastrophic cancellation' called so?
- Cardinality of a basis of an infinite-dimensional vector space
- Show that $q^4+2pq^2 +p^2 = 2pq -(pq)^2 -1$ becomes $p^3+q^3+3pq-1=0$.
- An analogue of Hensel's lifting for Fibonacci numbers