nLab separable metacompact spaces are Lindelöf


Recall that a topological space XX is metacompact if every open cover has a point-finite open refinement which covers XX. Further a space is called Lindelöf if every open cover has a countable sub-cover.


Assuming the law of excluded middle:

Every separable metacompact topological space XX is Lindelöf .


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 point finite refinement, which again must not be countable. Let DD\subset be a countable dense set. Then {U j} jJ\{U_j\}_{j\in J} is uncountable and so some point xDx \in D is in U jU_j for infinetly (even uncountably) many jJj \in J.



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