Intereting Posts

Show $\mathbb{Q}{2}]$ is a field by rationalizing
A normal intermediate subgroup in $B_3$ lattice?
Min-Max Principle $\lambda_n = \inf_{X \in \Phi_n(V)} \{ \sup_{u \in X} \rho(u) \}$ – Explanations
Anecdotes about famous mathematicians or physicists
Hartshorne II Ex 5.9(a) or R. Vakil Ex 15.4.D(b): Saturated modules
What groups can G/Z(G) be?
Prove that $Ω$ has no accumulation point
Sobolev meets Wiener
Number of vertices of a complete graph with $n$ edges
Can Peirce's Law be proven without contradiction?
Integral $S_\ell(r) = \int_0^{\pi}\int_{\phi}^{\pi}\frac{(1+ r \cos \psi)^{\ell+1}}{(1+ r \cos \phi)^\ell} \rm d\psi \ \rm d\phi $
How to compute the gcd of $x+a$ and $x+b$, where $a\neq b$?
Nested solutions of a quadratic equation.
General expression of $f(a, b)$ if $f(a, b)=f(a-1,b) + f(a, b-1) + f(a-1, b-1)$?
semigroup presentation and Diamond lemma

I want to prove the following:

Let $X$ be second countable zero-dimensional space. If $A,B \subseteq X$ are disjoint closed sets, there exist is a clopen set $C$ such that $A\subseteq C$ and $B\cap C = \emptyset $.

(A topological space $X$ is *zero-dimensional* if it is Hausdorff and has basis consisting of clopen sets.)

- If $L$ is a linear continuum in the order topology, then $L$ is connected.
- Separable implies second countable
- An example of topological space in which each singleton is not in $G_\delta$
- Can a continuous surjection from a Hilbert cube to a segment behave bad wrt Lebesgue measures?
- Decomposition of a manifold
- What does it mean to induce a topology?

- Every minimal Hausdorff space is H-closed
- sequential convergence and continuity
- Upper semicontinuous functions
- Compact spaces and closed sets (finite intersection property)
- Infinity-to-one function
- Continuously extending a set of independent vectors to a basis.
- manifold as simplicial complex
- Compact $G_\delta$ subsets of locally compact Hausdorff spaces
- Choosing a text for a First Course in Topology
- Complement is connected iff Connected components are Simply Connected

6.2.7. THEOREM.Every zero-dimensional Lindelöf space is strongly zero-dimensional.PROOF. It suffices to show that for every pair $A$, $B$ of disjoint closed subset of a zero-dimensional Lindelöf space $X$ there exists an open-and-closed set $U\subset X$ such that $A\subset U \subset X\setminus B$. For every $x\in X$ choose an open-and-closed set $W_x\subset X$ which contains $x$ and satisfies

$$A\cap W_x = \emptyset \qquad\text{or}\qquad B\cap W_x=\emptyset.$$

Let $\{W_{x_i}\}_{i=1}^\infty$ be a countable subcover of the cover $\{W_x\}_{x\in X}$ of the space $X$. The sets

$$U_i:=W_{x_i}\setminus \bigcup_{j<i} W_{x_i},\qquad\text{where }i=1,2,\dots$$

are open-and-closed and pairwise disjoint, and the family $\{U_i\}_{i=1}^\infty$ is a cover of the space $X$.

The set $U=\bigcup\{U_i : A\cap U_i \ne \emptyset\}$ has the required properties.

Source: Ryszard Engelking, *General Topology*, 2nd ed., Heldermann, Berlin, 1989.

- Motivation for construction of cross-product (Quaternions?)
- Fitting of Closed Curve in the Polar Coordinate.
- Sum of sets of measure zero
- For every cardinal $\kappa$, $\kappa^+$ is regular
- Method for coming up with consecutive integers not relatively prime to $(100!)$
- Solve a seemingly simple limit $\lim_{n\to\infty}\left(\frac{n-2}n\right)^{n^2}$
- how to show that a group is elementarily equivalent to the additive group of integers
- Evaluating the integral $\int_{-1}^{1} \frac{\sqrt{1-x^{2}}}{1+x^{2}} \, dx$ using a dumbbell-shaped contour
- Intuitive explanation of Left invariant Vector Field
- Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
- Nested Square Roots
- How many solutions are there to $x_1 + x_2 + … + x_5 = 21$?
- The only algebraic integers in $\mathbb Q $ are the ordinary integers
- Identity concerning complete elliptic integrals
- If $C$ is a component of $Y$ and a component of $Z$, is it a component of $Y\cup Z$?