Weakly Separated Spaces and Pixley–Roy Hyperspaces
Tóm tắt
In this paper we obtain new results regarding the chain conditions in the Pixley–Roy hyperspaces
$${\mathscr {F}\hspace{0mm}}[X]$$
. For example, if c(X) and R(X) denote the cellularity and weak separation number of X (see Sect. 4) and we define the cardinals
$$\begin{aligned} c^* (X):= \sup \{c(X^{n}): n\in {\mathbb {N}}\} \quad \text {and} \quad R^{*}(X):= \sup \{R(X^{n}): n\in {\mathbb {N}}\}, \end{aligned}$$
then we show that
$$R^{*}(X) = c^ {*}\left( {\mathscr {F}\hspace{0mm}}[X]\right) $$
. On the other hand, in Sakai (Topol Appl 159:3080–3088, 2012, Question 3.23, p. 3087) Sakai asked whether the fact that
$${\mathscr {F}\hspace{0mm}}[X]$$
is weakly Lindelöf implies that X is hereditarily separable and proved that if X is countably tight then the previous question has an affirmative answer. We shall expand Sakai’s result by proving that if
$${\mathscr {F}\hspace{0mm}}[X]$$
is weakly Lindelöf and X satisfies any of the following conditions:
then X is hereditarily separable.
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