Continuous Real-Valued Function on Irreducible Space is Constant

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Theorem

Let $T = \struct {S, \tau}$ be a topological space which is irreducible.

Let $f: S \to \R$ be a continuous real-valued function.


Then $f$ is constant, that is:

$\exists a \in \R: \forall x \in S: \map f x = a$


Proof

By definition of $T = \struct {S, \tau}$ being irreducible:

$\forall U_1, U_2 \in \tau: U_1, U_2 \ne \O\implies U_1 \cap U_2 \ne \O$


For every $x \in \R$, define:

$L_x := f^{-1} \sqbrk {\openint {-\infty} x}$
$U_x := f^{-1} \sqbrk {\openint x \infty}$

By continuity of $f$, these are open in $T$.

They are also disjoint, because for each $s \in S$, $\map f s < x$ and $\map f s > x$ cannot both hold simultaneously.


Suppose now that $\map f s = a$ and $\map f {s'} = b$, for some $s, s' \in S$ and $a, b \in \R$ with $a < b$.

Define $c = \dfrac{a + b} 2$.

Then $a < c < b$, whence:

$s \in L_c$ and $s' \in U_c$

However, because $T$ is irreducible, it follows that either $L_c$ or $U_c$ must be empty.


This contradicts the existence of such $a$ and $b$.

Hence $f$ must be constant.

$\blacksquare$


Sources