Property C � � and Function Spaces

The $\gamma _p$ property and the reals

“…These are generalizations of and actually imply the classical properties C and Hurewicz respectively; further, it is proved in [8] that a space X has property C (ω) if and only if each finite power X n has property C . Note that the γ F property defined above is clearly equivalent to the γ F property together with the Hurewicz(ω) property.…”

Combinatorics of Open Covers Vi: Selectors for Sequences of Dense Sets

“…In this proof we need to refer the reader to some facts from the literature -in particular [7], [13], [17], [18], [19] and [20]. The following theorems summarize some of the results we use.…”

“…Since C p (X) is homogeneous, countable strong fan tightness of C p (X) is equivalent to countable strong fan tightness at some f ∈ C p (X). In [18] Sakai proved for T 3 1 2 -spaces X that C p (X) has countable strong fan tightness if, and only if, X has property S 1 (Ω, Ω); in [20] I gave more characterizations for this, among others each of H 1 (Ω, Ω) for X and H 1 (Ω f , Ω f ) for C p (X) at some f is equivalent to the countable strong fan tightness of C p (X). Thus, we find from the results here that for X separable and metrizable, the countable strong fan tightness of C p (X) is equivalent to TWO not having a winning strategy in G D ω (C p (X)).…”

Countable size nonmetrizable spaces which are stratifiable and 𝜅-metrizable

“…A space X is said to have countable fan tightness [ 1 ] if for each x £ X and each countable system {An : n £ N} of subsets of X such that x £ f\ A" there exist finite sets B" of A" such that x £ (J Bn . If each B" is replaced with a singleton subset in the above definition, then X is said to have countable strong fan tightness [14].…”

“…A stratifiable K-metrizable space does not always have countable strong fan tightness. Since CP(C, {0, 1}) does not have countable strong fan tightness [14], there exists a sequence {An : n £ N} of subsets in CP(C, {0, 1}) such that fo £ f)A~n ; but for any h" £ A" f0 £ CP(C, {0, 1}) -{h" : « € N} . This implies that L in Example 2.2 does not have countable strong fan tightness.…”