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## Wydział Matematyki, Informatyki i Mechaniki Uniwersytetu Warszawskiego

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Sem. Topologia i T. Mnogości

## A universal coregular countable second-countable space

Prelegent: Taras Banakh

2021-05-26 16:15

A Hausdorﬀ topological space X is called superconnected (resp. coregular) if for any nonempty open sets U_1 , . . . ,U_n ⊆ X, the intersection of their closures cl(U_1)∩...∩cl(U_n) is not empty (resp. the complement X \ (cl(U_1)∩...∩cl(U_n)) is a regular topological space). A canonical example of a coregular superconnected space is the projective space QP^∞ of the topological vector space Q^<ω = {(x_n)_{n∈ω} ∈ Q^ω : |{n ∈ ω : x_n \neq 0}| < ω} over the ﬁeld of rationals Q. The space QP^∞ is the quotient space of Q^<ω \ {0}^ω by the equivalence relation x ∼ y iﬀ Q·x = Q·y. We prove that every countable second-countable coregular space is homeomorphic to a subspace of QP^∞ , and a topological space X is homeomorphic to QP^∞ if and only if X is countable, second-countable, and admits a decreasing sequence of closed sets (X_n )_{n∈ω} such that (i) X_0 = X, \bigcap_{n∈ω}X_n = ∅, (ii) for every n ∈ ω and a nonempty open set U ⊆ X_n the closure cl(U) contains some set X_m, and (iii) for every n ∈ ω the complement X \ X_n is a regular topological space. Using this topological characterization of QP^∞ we ﬁnd topological copies of the space QP^∞ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological ﬁelds.