Suprema of continuous functions on connected spaces

Abstract

Let $K$ be a compact Hausdorff space and let $(f_n)_{n\in \N}$ be a pairwise disjoint sequence of continuous functions from $K$ into $[0,1]$. We say that a compact space $L$ \emph{adds supremum} of $(f_n)_{n\in \N}$ in $K$ if there exists a continuous surjection $\pi:L\longrightarrow K$ such that there exists $sup\{f_n\circ\pi:n\in \N\}$ in $C(L)$. Moreover, we expect that $L$ preserves suprema of disjoint continuous functions which already existed in $C(K)$. Namely, if $sup\{g_n:n\in \N\}$ exists in $C(K)$, we must have $sup\{g_n\circ\pi:n\in \N\}$ in $C(L)$. This paper studies the preservation of connectedness in extensions by continuous functions -- a technique developed by Piotr Koszmider to add suprema of continuous functions on Hausdorff connected compact spaces -- proving the following results: (1) If $K$ is a metrizable and locally connected compactum, then any extension of $K$ by continuous functions is connected (but it may be not locally connected). (2) There exists a disconnected extension of a metrizable connected compactum $K$. (3) For any metrizable compactum $K$ there exists a disconnected $L$ which is obtained from $K$ by finitely many extensions by continuous functions.