nLab
paracompact spaces satisfying the countable chain condition are Lindelöf

Statement

Recall that a topological space XX is paracompact if every open cover has a refinement by a locally finite open cover. Further a space is called Lindelöf if every open cover has a countable sub-cover.

Theorem

Assuming the axiom of choice:

Every paracompact space XX satisfying the countable chain condition is Lindelöf.

Proof

The proof goes by contradiction: Assume there is an open cover {U i} iI\{U_i\}_{i\in I} with no countable subcover. Let {U j} jJ\{U_j\}_{j\in J} be a locally finite refinement, which again must not be countable. This is to say that each point possesses an open neighborhood so small that it is only an element of finitely many U jU_j‘s. Inductively (using Zorn's lemma) construct a maximal system {V λ} λΛ\{V_\lambda\}_{\lambda \in \Lambda} of pairwise disjoint opens being contained in at most finitely many U jU_j’s. Due to maximality λΛV λ\bigcup_{\lambda\in\Lambda} V_\lambda is dense. This fact implies by the countable chain condition that Λ\Lambda is countable. Moreover it implies that each U jU_j intersects at least one V λV_\lambda. But this is to say that there are at most countably many U jU_j’s. This is a contradiction.

Properties

Implications

Last revised on April 3, 2020 at 15:37:40. See the history of this page for a list of all contributions to it.