Order types, calibres and spread of Corson compacta (Q1183668)

For a set \(\Gamma\), let \(\Sigma(\Gamma)\) be the \(\Sigma\)-product of the real line; i.e. \(\Sigma(\Gamma)=\{x\in R: \text{supp}(x)=\{\gamma\in \Gamma: x(\gamma)\neq 0\}\) is countable\}. A Corson compactum is a compact subspace of \(\Sigma(\Gamma)\) for some set \(\Gamma\). For a 0-dimensional Corson compactum \(K\), order type of \(K\) denotes the least ordinal \(\alpha\) such that there are an ordinal \(\eta\) and an embedding \(h: K\to\Sigma(\eta)\cap\{0,1\}^ \eta\) such that order type of \(\text{supp}(h(k))\leq\alpha\) for all \(k\in K\). For a topological space \(X\), \(s(X)\) denotes the spread of \(X=\sup\{| Y|: Y\subset X\) is discrete\} and \(w(X)\) denotes the weight of \(X\). The author discusses the situation of cardinality functions mentioned above and obtains the following results: 1. Zero-dimensional Eberlein compacta have order type \(\omega\), and there is a zero-dimensional Talagrand compactum with order type \(\omega_ 1\). 2. Assume that there exists a Souslin line, i.e. a ccc nonseparable linearly ordered space. Then there exists a Corson compactum \(K\) with \(s(K)=\omega\) and \(w(K)=\omega_ 1\). 3. Let \(\Gamma\) be a set and \(X\subset\Sigma(\Gamma)\). If \(w(X)>\omega_ 1\), then \(s(X)=w(X)\). Therefore, if \(K\) is a Corson compactum with \(w(K)>\omega_ 1\) or \(K\) is a Gul'ko compactum, then \(s(K)=w(K)\). 4. If \(K\) is a Corson compactum of order type \(<\omega_ 1\), then \(s(K)=w(K)\). The relationship between those classes of compact spaces is as below: \[ \text{Eberlein}\Longrightarrow \text{Talagrand}\Longrightarrow \text{Gul'kov}\Longrightarrow \text{Corson}. \] .