Condensations of Tychonoff universal topological algebras.

*(English)*Zbl 1053.54044The author adjusts a general construction due to D. B. Shakhmatov to obtain a result, in some sense similar to the well-known result of A. V. Arhkangel’skij: Every Hausdorff topological group \(G\) of network weight \(\leq \tau \) can be condensed isomorphically onto a Hausdorff topological group \(G^*\) of weight \(\leq \tau \). There is a key and very natural question, whether one can replace the group structure on \(G\) by some other, more general structure, and the Hausdorff axiom by some stronger (regular of Tychonoff) separation axiom. Let us note that all considered spaces are at least Hausdorff. To formulate the main result, the following notion is needed: Let \(X\) be a space, \(\mathcal K\) a class whose elements are external to \(X\); and the spaces \(K_1,\dots , K_m\) are in \(\mathcal K\). An \((n,m)\)-ary continuous operation on \(X\) is a pair \((j, D_j)\) consisting of the domain \(D_j\subseteq X^n\times K_1\times \dots \times K_m\) of the operation \(j\) and of a continuous mapping: \(j: D_j\rightarrow X\). The set \(D\) is considered with the topology induced from \(X^n\times K_1\times \dots \times K_m\). The following theorem is the author’s main result: Let \((X,\mathcal T)\) be a (Tychonoff) regular space and \(\mathcal K\) be a class of topological spaces. Suppose that there are specified \(\leq \tau \) continuous operations on the space \((X,\mathcal T)\). If \(nw(X,\mathcal T)\leq \tau \) and all \(K\in \mathcal K\) satisfy \(nw(K)\leq \tau \), then there exists a condensation \(i:(X,\mathcal T)\rightarrow (X,\mathcal T^*)\) where \(\mathcal T^*\) is a coarser (Tychonoff) regular topology on \(X\) such that all operations remain continuous on the space \((X,\mathcal T^*)\) and \(w(X,\mathcal T) \leq \tau \). The author closes the paper by an example of a Hausdorff but non-regular paratopological group.

Reviewer: Martin Kovár (Brno)