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Random Talk on Random Thoughts

Mountain Climbing in Lindelöf $T_3$ Spaces

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To deduce that compact $T_2$ spaces are $T_4$, I tried using the arguments of the proof in the previous post, but I couldn’t do it. Therefore, I read another proof about the normality of regular Lindelöf spaces instead.

The proof at first glance

Suppose that $X$ is the given space, and $H$ and $K$ are the given sets. $\left\{G_n^* : n \in \omega\right\}$ and $\left\{W_n^* : n \in \omega\right\}$ are open countable subcovers of $H$ and $K$ respectively. On p.91 in Davis's Topology book, I saw the following equations.

\begin{alignat*}{2} U_0 &= G_0^* &\quad V_0 &= W_0^* \setminus \overline{U_0} \\ U_1 &= G_1^* \setminus \overline{V_0} &\quad V_1 &= W_1^* \setminus (\overline{U_0} \cup \overline{U_1}) \\ U_2 &= G_2^* \setminus (\overline{V_0} \cup \overline{V_1}) &\quad V_2 &= W_2^* \setminus (\overline{U_0} \cup \overline{U_1} \cup \overline{U_2}) \\ \vdots \\ U_n &= G_n^* \setminus \bigcup_{k < n} \overline{V_k} &\quad V_n &= W_n^* \setminus \bigcup_{k \le n} \overline{U_k} \\ U &= \bigcup_{n = 0}^\infty U_n &\quad V &= \bigcup_{n = 0}^\infty V_n \end{alignat*}

I didn’t know what to do when I first saw these equalities. It’s hard to understand the reason of $U \cap V = \varnothing$ by just reading the symbols. Luckily, an analogy of climbing mounts is given. Therefore, I finally understood how the open subsets $U_n$’s and $V_n$’s are built step-by-step while preserving disjointness and openness.