Suprema of continuous functions on connected spaces

Abstract

Let K be a compact Hausdorff space and let (f n ) n∈N be a pairwise disjoint sequence of continuous functions from K into [0, 1]. We say that a compact space L adds supremum of (f n ) n∈N in K if there exists a continuous surjection π : L −→ K such that there exists sup{f n • π : n ∈ 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 ∈ N} exists in C(K), we must haveThis 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.

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“…Items (6) - (7) are the immediate consequences of (4) and the properties of the preimages of functions. For the forward direction of (8), note that always π −1 F ,G [U (F )] ⊆ π −1 F ,G [U (F )] and apply (4). For the backward direction of (8), note that always π F ,G [U (G)] = π F ,G [U (G)] by Lemma 3.9 and apply (5).…”

“…As F r(κ) is a dense subalgebra of F r(κ), it is enough to prove that p A [s F r(κ) (a)] has a nonempty interior in L A for any a ∈ F r(κ) (see section 2. Indiscriminate adding of suprema leads to a complete lattice C(K) and implies that K is extremally disconnected, so in general extensions of compact spaces do not need to preserve the connectedness (for explicite analysis of this phenomenon in the case of pairwise disjoint sequences of functions see [4]), however we have the following: Proof. Use 5.13. of [24] and Lemma 3.12.…”

“…Indiscriminate adding of suprema leads to a complete lattice C(K) and implies that K is extremally disconnected, so in general extensions of compact spaces do not need to preserve the connectedness (for explicite analysis of this phenomenon in the case of pairwise disjoint sequences of functions see [4]), however we have the following: Lemma 3.27 ([21], 4.4. ).…”