Intereting Posts

Radius of circumscribed circle of triangle as function of the sides
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Cardinality of Cartesian Product of finite sets.
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Hall's theorem vs Axiom of Choice?
Let $f$ be a continuous function satisfying $\lim \limits_{n \to \infty}f(x+n) = \infty$ for all $x$. Does $f$ satisfy $f(x) \to \infty$?
Difference between the algebraic and topological dual of a topological vector space?
a theorem about converge pointwisely and uniformly
Prove that $f$ is a polynomial if one of the coefficients in its Taylor expansion is 0
What is the angle between those two matrices over $\mathbb{C}$?
Find all rings $R$ satisfying $\mathbb Z \subset R \subset \mathbb Z$
The correspondence theorem for groups
Suppose $A$ is a 4×4 matrix such that $\det(A)=\frac{1}{64}$
Simplifying Ramanujan-type Nested Radicals

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.)

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- Example of Hausdorff and Second Countable Space that is Not Metrizable
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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.

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