Suprema of continuous functions on connected spaces

TL;DR

This paper investigates how extending connected compact spaces with continuous functions affects their connectedness, showing conditions under which connectedness is preserved or lost in such extensions.

Contribution

It proves that extensions of metrizable, locally connected compacta remain connected, but disconnected extensions can also be constructed, highlighting the nuanced effects of these extensions.

Findings

Extensions of metrizable, locally connected compacta are connected

Disconnected extensions of connected compacta exist

Finite extensions can produce disconnected spaces from connected compacta

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.