On cardinality bounds for \(\theta^n\)-Urysohn spaces (Q2330060)

The authors define the notion of \(n\)-\(\theta\)-closure: for a subset \(A\) of a space \(X\), \(cl^n_{\theta}(A) =cl_{\theta}\cdots cl_{\theta}(A)\) (the operator \(cl_{\theta}\) taken \(n\)-times). Using this operator, they then define \(\theta^n\)-Urysohn spaces as spaces in which any two distinct points have neighbourhoods with disjoint \(n\)-\(\theta\)-closures. A few cardinal functions (as the \(n\)-\(\theta\)-almost Lindelöf degree \(\theta^n\)-\(aL(X)\) and the \(\theta^n\)-Urysohn cellularity \(\theta^n\)-\(Uc(X)\)) of a space \(X\) are defined and used to extend to \(\theta^n\)-Urysohn spaces some cardinal inequalities known for Urysohn spaces. Sample results are: (1) If \(X\) is a \(\theta^n\)-Urysohn space, then \(\vert X\vert \le 2^{\theta^n\!\!-Uc(X)\chi(X)}\) and \(\vert X\vert \le 2^{\theta^n\!\!-aL(X)\chi(X)}\); (2) if \(X\) is a homogeneous \(\theta^n\)-Urysohn space, then \(\vert X\vert \le 2^{\theta^n\!\!-Uc(X)\pi\chi(X)}\).