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?

Wikipedia describes the process of creating the Cantor set at https://en.wikipedia.org/wiki/Cantor_set We begin with one segment $[0,1]$ and remove $ \left( \frac{1}{3},\frac{2}{3} \right)$ This leaves us with two segments $\left[ 0,\frac{1}{3}\right] \left[ \frac{2}{3},1\right]$ and remove $ \left( \frac{1}{9},\frac{2}{9} \right) \left( \frac{7}{9},\frac{8}{9} \right)$ This leaves us with four segments $\left[ 0,\frac{1}{9}\right] \left[ \frac{2}{9},\frac{1}{3}\right] \left[ \frac{2}{3},\frac{7}{9}\right] \left[ […]

Is it true that if $F$ is a locally compact topological field with a proper nonarchimedean absolute value $A$, then $F$ is totally disconnected? I am aware of the classifications of local fields, but I can’t think of a way to prove this directly.

In other words does there exist a non discrete metric space in which Closed balls are subset of the closure of open balls?

This is homework. I need to show that every AR is contractible. All I can basically do here is list definitions: A space $Y$ is AR if: $X$ is metrizable, $A$ is closed subset of $X$ and $f: A \mapsto Y$ is continuous, then $f$ has a continuous extension $g: X \mapsto Y$. A space […]

So I have been reading through some threads on Mathstack about the irreducible components of a topological space, and it suddenly dawned on me: How can we guarantee that any non-empty topological space actually contains an irreducible closed subset? Ok so on the surface it would appear to make sense, as if it didnt contain […]

I have a general outline of the proof of the Arzela-Ascoli Theorem but have trouble filling in the gaps of the theorem. I have posted the entire general method I believe to be correct below. I was wondering if there was an easier way to do this or if someone would know of an easier […]

I have seen the proof of (countable complement topology is not meta compact) , which says that the countable intersection of open sets is open and thus uncountable, so this topology cannot be meta compact. It is easy to see that the countable intersection is open and uncountable, but why this implies that this topology […]

What I am trying to understand is what is going on with the inclusion of sets, as if I understand correctly they are the morphisms of the category of open sets on X: $Op_X$ is the category of open sets such that for open $U,V \in X$: $\ Hom_{Op_X}(U,V) = \begin{cases} \{pt\} & U \subset […]

Let $S$ be a subset of $\mathbb{R}^n$; show that the set $I$ of isolated points of $S$ is countable. Let $\mathbf{x}\in I$. There exists an open ball, say $B(\mathbf{x},r_\mathbf{x})$, of radius $r_\mathbf{x}$, for each $\mathbf{x}$ such that $B(\mathbf{x},r_\mathbf{x}) \cap S=${$\mathbf{x}$}. Now without change of notation replace $ r_\mathbf{x}$ with $\frac{1}{2}r_\mathbf{x}$. Thus the open balls $B$ […]

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