Classification of bicompacta starting from Eberlein bicompacta (Q1173555)

For a space \(X\) let \(t(X)\), \(\omega(X)\) and \(d(X)\) denote the tightness, the weight and the density, respectively. Let \(C_ p(X,Y)\) be the space of all continuous maps \(f: X\to Y\) with the topology of pointwise convergence and set \(C_ p(X) = C_ p(X,\mathbb{R})\). If \(X\) is a bicompact space and \(Y \subset C_ p(X)\) separates points of \(X\), then the mapping \(\psi_ Y: X\to C_ p(Y)\), defined by \(\psi_ Y(x)(g) = g(x)\) for each \(g\in Y\), is a homeomorphism such that \(\psi_ Y(x)\) separates the points of \(Y\). Similarly, the author proves that if \(X\) is a bicompact space and \(F \subset C_ p(X,\mathbb{R}^ \tau)\) separates points of \(Y\), then the mapping \(\psi_ F: X\to C_ p(F,\mathbb{R}^ \tau)\), defined by \(\psi_ F(x)(f) = f(x)\) for each \(f \in F\), is a homeomorphism such that \(\psi_ F(X)\) separates points of \(F\). The author introduces the notions of Eberlein number and Corson number as follows: For each bicompact space \(F\) the number \(\aleph_ 0 \cdot \min\{\tau:\) there is a \(T_ 0\)-separating family \(\gamma = \bigcup\{\gamma_ \alpha: \alpha < \tau\}\) in \(F\) of point-finite families \(\gamma_ \alpha\) consisting of cozero-sets\} is called the Eberlein number of the space \(F\) and is denoted by \(\text{eber}(F)\). For each bicompact space \(X\) the number \(\aleph_ 0 \cdot \min\{\tau: X\) is embeddable \(\Sigma_ \tau(A)\) \((\Sigma_ \tau(A) = \{x \in \mathbb{R}^{| A|}: |\{\alpha \in A: x_ \alpha \neq 0\}| \leq \tau\}\) for some set \(A\}\) is called the Corson number of the space \(F\) and is denoted by \(\text{cor}(F)\). It is proved that these notions have the following properties: For each bicompact space \(X\), \(t(X) \leq \text{cor}(X) \leq \text{eber}(Y) \leq \omega(X)\); for the product \(X = \prod\{X_ \alpha: \alpha \in A\}\) of bicompact spaces \(X_ \alpha\) which contain more than one point the following hold: (a) \(\text{cor}(X) = | A|\text{sup}\{\text{cor}(X_ \alpha): \alpha \in A\}\), (b) \(\text{eber}(X) = | A|\text{sup}\{\text{eber}(X_ \alpha): \alpha\in A\}\).