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**CSM** I’m having a really hard time with this one. On a conceptual level and on a technical level. I’ve got my three topology books here, but I can’t seem to make anything out of it. Can somebody enlighten me?

Let $\displaystyle T$ be a topological space. And let $\displaystyle T$ be normal and $\displaystyle \{C_1, C_2,…,\}$ a countable family of closed subsets of $\displaystyle T$. Suppose that every point of $\displaystyle T$ has an neighbourhood $\displaystyle U$ such that $\displaystyle U\cap C_i\not=\emptyset$ for at most one $\displaystyle C_i$. (Note in perticular that $\displaystyle C_i\cap C_j=\emptyset$ when $\displaystyle i\not= j$.)

Prove: there are open sets $\displaystyle U_1,U_2,… $ such that $\displaystyle C_i\subset U_i$ for each $\displaystyle i$ and such that $\displaystyle \overline{U_i}\cap\overline{U_j}=\emptyset$ when $\displaystyle i\not= j$

(Use Tietze Extension Theorem or Urysohn’s lemma)