A note on chains of open sets

Abstract:

We consider some questions concerning the nature and size of chains of open sets in Hausdorff spaces. The following results are obtained. Theorem 1. For every cardinal $\kappa$ there exists a space $X$ in which all discrete subsets have cardinality at most $\kappa$ and which contains a chain of ${({2^\kappa })^ + }$ open sets. Theorem 2. If $X$ is regular and contains a chain of ${({2^\kappa })^ + }$ open sets, then $X \times X$ contains a discrete subset of cardinality ${\kappa ^ + }$. Theorem 3. Let $M(X)$ denote the set of all maximal chains of open subsets of $X$ endowed with the Tychonoff topology. (i) $\left | {M(X)} \right | \leqslant {2^{{\text {w}}(X)}}$, and (ii) $\psi (M(X)) \leqslant {\text {w}}(X)$. Here ${\text {w}}(X)$ denotes the weight of the space $X$ and $\psi (M(X))$ denotes the pseudocharacter of the space $M(X)$.