Every topology has a bijection with the Hom-Set of its continuous functions to the Sierpiński space

up:: Sierpiński Space

Given a topological space $(Z,\tau )$, one can create a Bijection between $\tau$ and the Hom-Set of all Topologically Continuous Functions $Hom(Z,S)$.

Given any other topological space $(Z,\tau )$, for each open set $U\in \tau$, define $F:\tau \to Hom(Z,S)$ such that $F(U):={\chi }_U$ is the characteristic function of this open set $U$ onto the Sierpiński space $(S,{\tau }_S)$:

$$\begin{array}{rl}{\chi }_U:& Z\to S\\ & z\mapsto \{\begin{array}{l}1,z\in U\\ 0,z\notin U\end{array}\end{array}$$

Note that it is continuous, since, for every open set in ${\tau }_S$,

$$\begin{array}{rl}{\chi }_U^{-1}(\{0,1\})& =Z\in \tau \\ {\chi }_U^{-1}(\{1\})& =U\in \tau \\ {\chi }_U^{-1}(\varnothing )& =\varnothing \in \tau \end{array}$$

Note that $F$ is Injective, since different sets $U,V$ will necessarily have different characteristic functions.

Conversely, note that, for any continuous function $f:Z\to S$, it will have only one open set $U_f=f^{-1}(\{1\})\in \tau$ associated to it. Note that this mapping $f\mapsto U_f$ is injective as well, else their preimages would be the same.

Thus, through the Cantor-Bernstein-Schroeder Theorem, we conclude that there is a bijection between $\tau$ and $Hom(Z,S)$, where the open set $\{1\}\in {\tau }_S$ acts as the “indicator” for each open set $U_f\in \tau$ under the continuous function $f:Z\to S$.

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References

• The Sierpinski Space and Its Special Property