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- W2277394430 abstract "Let $K$ be a compact Hausdorff space and let $(f_n)_{nin N}$ be a pairwise disjoint sequence of continuous functions from $K$ into $[0,1]$. We say that a compact space $L$ emph{adds supremum} of $(f_n)_{nin N}$ in $K$ if there exists a continuous surjection $pi:Llongrightarrow K$ such that there exists $sup{f_ncircpi:nin N}$ in $C(L)$. Moreover, we expect that $L$ preserves suprema of disjoint continuous functions which already existed in $C(K)$. Namely, if $sup{g_n:nin N}$ exists in $C(K)$, we must have $sup{g_ncircpi:nin N}$ 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." @default.
- W2277394430 created "2016-06-24" @default.
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- W2277394430 date "2016-02-22" @default.
- W2277394430 modified "2023-09-23" @default.
- W2277394430 title "Suprema of continuous functions on connected spaces" @default.
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