Articles of general topology

Irreducible set

Let $X$ be a topological space Let $A$ be the set of all closed, irreducible subsets of $X$ equipped with a topology that contains all sets of the form $V(U)=\{a\in A| a\cap U\neq\emptyset, \text{where $U\in$ (the topology of $X$, $T_X$)}\}$. So the topology of $A$ is $\{V(U)|U\in T_X\}$ What then is a closed, irreducible subset […]

“Truncated” metric equivalence

I may be having problems with this proof. Consider the metric on a set $S$ defined by: $$d^\prime (x,y) = \left\{ \begin{array}{lcr} d(x,y) & \text{if}& d(x,y) \leq 1\\ 1 & \text{if} & d(x,y) > 1 \end{array} \right.$$ where $d$ is a metric on $S$. I’m trying to determine whether or not $d^\prime$ and $d$ induce […]

Distance between a point and a closed set in metric space

Here is what I am thinking. Let (X,d) be a metric space and let C be a closed subset of X. Fix any poin p in X. Then, there exists a point q in C such that d(p,q) = distance(p,C). I think this statement is true, so I tried to the following proof. For any […]

Quotient space and Retractions

I’m trying to learn something about topology and category theory. Let us consider the category of compact Polish spaces. The category contains all quotients of all objects (wikipedia) For an equivalence relation $\sim$ on $X\times X$, we denote with $X/\!\sim$ the quotient space and with $q:X\rightarrow X/\!\sim$ the quotient map. I’m trying to convince myself […]

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 […]