Effective Presentations of Mathematical Structures

Abstract

<p dir="ltr">For several algebraic structures, we show that they have a punctual dimension either 1 or ∞. The punctual degree structures of rigid structures and the ordered integers are dense. We consider some classes where every punctual structure from the class can be punctually embedded into its punctual (existential, algebraic) closure. We prove that the space C[0, 1] and the Urysohn space U are computably and punctually universal, and that the Urysohn space is not punctually categorical. And finally, we show that effectively compact punctual presentations of a Stone space are punctually homeomorphically embeddable into Cantor space, and that there is a compact totally disconnected punctual Polish space which is not computably homeomorphically embeddable into Cantor space.</p>