Articles of general topology

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

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?

How many segments are there in the Cantor set?

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[ […]

Locally compact nonarchimedian fields

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.

Does there any non discrete metric space in which closure of an open ball is not closed ball?

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

Every absolute retract (AR) is contractible

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 […]

Does every non-empty topological space have an irreducible closed subset?

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 […]

Gaps in my proof of the Arzela-Ascoli Theorem – help and expertise greatly appreciated for an alternate formulation.

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 […]

Prove that the countable complement topology is not meta compact?

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 […]

Understanding the inclusion of sets in the open category of X $Op_X$ and what \{pt\} denotes

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 […]

Show that the set of isolated points of $S$ is countable

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 $B$ […]