Embeddings of bornological universes (Q1005120)

A bornological universe \(\left\langle X,\tau,\mathcal B\right\rangle\) is a topological space \(\left\langle X,\tau\right\rangle\) equipped with a bornology \(\mathcal B\), that is, a cover of \(X\) that is hereditary and is closed under finite unions. The author proves that the space \(X\) can be topologically and bornologically embedded in \(\mathbb{R}^Y\) for some index set \(Y\) if and only if one of the following three conditions holds: (1)~for each nonempty \(B\in\mathcal B\) there exists \(f\in C(X,[0,1))\) such that \(f(B)=0\) and \(f^{-1}([0,1))\in\mathcal B\) and there exists a countable subset of \(\mathcal B\) forming a cover of~\(X\); (2)~the one-point extension \(o(\mathcal B)\) of \(\left\langle X,\tau\right\rangle\) is Tychonoff and the ideal point is a \(G_{\delta}\)-subset of the extension; (3)~for each \(B\in\mathcal B\) there exists a continuous unbounded function \(f:X\to[0,\infty)\) such that \(f^{-1}([0,\alpha))\in\mathcal B\) for each \(\alpha>0\) and \(f(B)=0\). The special case when the bornology has a countable base is studied with more detail.