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**Lemma:**(Professor said we would be able to use this for the problem? But we have to prove it to use it.)

If X is a regular Hausdorff space and X is first countable at the point P. Then there is a local basis $\left\{B_i\right\}^{\infty}_{i=1}$ at P so that for each $n \in \Bbb{N}$ we have:

$\overline{B_{n+1}} \subset B_n$.

**Proof of Lemma:**

X first countable at P $\implies \exists$ a countable local basis $\left\{A_i\right\}^{\infty}_{i=1}$ at P.

Let $B_n(P) = A_1 \cap A_2 \cap A_3 \cap ... \cap A_n$.

Then $\left\{B_i(P)\right\}^{\infty}_{i=1}$ is a collection of open sets such that

$B_{i+1}(P) \subset B_i(P)$, for all i.

*Need to Show:*$\left\{B_i(P)\right\}^{\infty}_{i=1}$ is a local basis at P.

Consider an open set B containing P.

$\left\{A_i\right\}^{\infty}_{i=1}$ is a local basis at P and B is an open set containing P $\implies$ (by first countability)

$\exists i_0 \in \Bbb{N}$ such that $A_{i_0} \subseteq B$.

By definition of the $B_n(p)$'s,

$A_1 \cap A_2 \cap ... \cap A_{i_0} = B_{i_0}(p) \subseteq A_{i_0} \subseteq B$

So for any open set B containing P, $\exists i_0 \in \Bbb{N}$ so that $B_{i_0}(P) \subseteq B$.

$\implies \left\{B_i(P)\right\}^{\infty}_{i=1}$ is a local basis at P.

*Need to Show:*for each $n \in \Bbb{N}, \overline{B_{n+1}} \subset B_n$

X regular $\implies$ if $P \in B_i(P) \implies \exists U$ open in X such that $P \in U \subset \overline{U} \subset B_i(P)$

However, $\left\{B_i(P)\right\}^{\infty}_{i=1}$ is a local basis at P $\implies$

$\exists B_m(P)$ so that $P \in B_m(P) \subset U$

$\implies P \in B_m(P) \subset \overline{B_m(P)} \subset U \subset B_i(P)$

Then, $\overline{B_m(P)} \subset B_i(P)$.

**I'm sort of stuck here... is this the right idea for the proof or not so much?**

**Problem 1.**

Suppose that X is a linearly ordered space that has first and last points a and b respectively. Suppose further that X is first countable at both endpoints. Show that there is a continuous function $f : X \rightarrow [0, 1]$ so that $f^{−1}(0) = {a}$ and $f^{−1}(1) = {b}$.

**Proof:**

I chose the function

$f(x) = \frac{x - a}{b - a}$

**I think this should work fine. Do I just need to prove continuity now?**