An exercise on the characteristic function and the Sierpinski-space

5 Juin 20234 Juin 2023 / Christian Nguembou Tagne

The characteristic function appears in many fields of mathematics. This publication set an exercise that allows to discover elementary features of the characteristic function in the context of Set Theory and General Topology. The exercise also introduces the Sierpinski-space.

In a preceding entry of this blog, we established that there exactly four topologies on a two-elements set: the indiscrete topology, the discrete topology, and two equivalent topologies. The Sierpinski-space is a two-elements set endowed with one of these equivalent topologies.

Exercise.

We set $2=\bigl\{0,1\bigr\}$ , and recall that the characteristic function of a subset $A$ of a set $X$ is the function denoted by $\chi_{A}$ , from $X$ into $2$ , defined by

$\chi_{A}(x)=\left\{\begin{array}{l} 1\,\text{ if }\, x\in A, \\[6pt] 0\,\text{ if }\, x\in X\setminus A.\end{array}\right.$

Verify that the function $\gamma:\mathcal{P}(X)\rightarrow 2^{X}$ , from the power set of $X$ into the set of all functions of $X$ into $2$ , given by $\gamma(A)=\chi_{A}$ , is a bijection.
Given a set $X$ and a subset $A$ of $X$ , show that the set $\mathfrak{T}=\bigl\{\emptyset, A,X\bigr\}$ is a topology on $X$ . In particular, the set $\mathfrak{A}=\bigl\{\emptyset,\{1\},2\bigr\}$ is a topology on $2$ . The topological space $(2,\mathfrak{A})$ is called the Sierpinski-space.
Let $(X,\mathfrak{O})$ be a topological space. Prove that the characteristic function $\chi_{A}:(X,\mathfrak{O})\rightarrow(2,\mathfrak{A})$ of a subset $A$ of $X$ is continuous if, and only if, $A\in\mathfrak{O}$ .