Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions

Question

Show that lower semicontinuous function $f:X\rightarrow [0,1]$ on metrizable X is the supremum of an increasing sequence of continuous functions.

My attempt: I don't know how to approximate $f(x)$ to within $[f(x)-1/n,f(x)]$ by $h_n(x)$ which is a linear combination of characteristic function of open sets, using lower semicontinuity of $f$.

Can anyone help?

Answer 1

This is a quotation from "General Topology" by Ryszard Engelking:

Answer 2

This is a problem from Willard's general topology textbook. I shall provide an outline of the proof and leave the details to the reader.
The function $f(x)$ may be approximated to within $[f(x)-1/n, f(x)]$ by the function $$h_{n} = \frac{1}{n}\sum_{k = 1}^{n-1}\chi_{f^{-1}\left(\left(\frac{k}{n}, 1\right]\right)}$$ where $\chi_{A}$ denotes the characteristic function of a subset $A$ of $X$. Note that $h_{n}$ is a linear combination of characteristic functions of open sets.
The characteristic function of any open set $A$ in a metric space $(X, \rho)$ can be written as the supremum of an increasing sequence of continuous functions. To prove this, note that every open set in a metric space is $F_{\sigma}$. In particular, if $A$ is an open set then $A$ can be written as the union of an increasing sequence $A_{1}\subseteq A_{2}\subseteq\cdots$ of closed sets where $$A_{n} = \left\{y\in X: \rho(y, X-A)\geq \frac{1}{n}\right\}.$$ For each positive integer $n$ define $g_{n}:X\rightarrow\mathbb{R}$ by $g_{n}(x) = \min\{n\rho(x, X-A), 1\}$ and verify that $\chi_{A} = \sup_{n}g_{n}$.
For each $1\leq k\leq n-1$ let $$\frac{1}{n}\chi_{f^{-1}\left(\left(\frac{k}{n}, 1\right]\right)} = \sup_{m}g_{k,m}$$ for an increasing sequence of continuous functions $\{g_{k, m}\}_{m\in\mathbb{Z}^{+}}$. One can verify that $h_{n} = \sup_{m} h_{n, m}$ where $h_{n, m} = \sum_{k = 1}^{n-1}g_{k,m}$.
Finally, one can show that $$f = \sup_{(n, m)\in\mathbb{Z}^{+}\times\mathbb{Z}^{+}}h_{n, m}.$$ Using any bijection between $\mathbb{Z}^{+}\times\mathbb{Z}^{+}$ and $\mathbb{Z}^{+}$, rearrange the functions $\{h_{n,m}\}$ into a sequence $\{\tilde{k}_{n}\}$ and let $k_{n} = \max\{\tilde{k}_{1},\dots,\tilde{k}_{n}\}$. Then $\{k_{n}\}$ is the required sequence of increasing continuous functions such that $f = \sup_{n}k_{n}$.