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By James C. Alexander, John L. Harer

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Nagami, Countable paracompactness of inverse limits and products, Fund. Math. 73 (1972), no. 3, 261–270. [26] N. Noble, Countably compact and pseudo-compact products, Czechoslovak Math. J. 19(94) (1969), 390–397. [27] H. Ohta, Topologically complete spaces and perfect maps, Tsukuba J. Math. 1 (1977), 77–89. [28] H. Ohta, Extension properties and the Niemytzki plane, Appl. Gen. Topol. 1 (2000), no. 1, 45–60. [29] H. Ohta and K. Yamazaki, Extension of point-finite partitions of unity, Fund. Math.

For products of weak topologies, see [80, 81]. For a space X with a point-countable determining cover by cosmic spaces, χ(X) ≤ 2c , c = 2ω . But, for each α ≥ ω, there is a symmetric space X with a point-finite determining closed cover by metric spaces such that χ(X) > α, and χ(X) > c when we replace “metric spaces” by “compact metric spaces”. Let X 2A space X is a Tanaka space if for a decreasing sequence {A : n ∈ N} with x ∈ A \ {x} n n for all n ∈ N, there exist xn ∈ An such that the sequence {xn : n ∈ N} converges to some point in X.

Problem 10. Is it true that every LΣ(<ω)-space is a union of countably many subspaces that admit continuous bijections onto second-countable spaces? 141? It is proved in [3] that LΣ(≤ω)-spaces have no uncountable free sequences; in particular, every compact LΣ(≤ω)-space has countable tightness. On the other hand, [3] contains an example that shows that σ-compact LΣ(<ω)-spaces may have arbitrary tightness. Problem 11. Let X be an LΣ(n)-space for some n ∈ ω. Can X have uncountable tightness? 142?

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