Point-set topology as diagram chasing computations: Lifting property as negation

Abstract: We observe that some natural mathematical definitions are lifting properties relative to simplest counterexamples, namely the definitions of surjectivity and injectivity of maps, as well as of being connected, separation axioms $T_0$ and $T_1$ in topology, having dense image, induced (pullback) topology, and every real-valued function being bounded (on a connected domain). We also offer a couple of brief speculations on cognitive and AI aspects of this observation, particularly that in point-set topology some arguments read as diagram chasing computations with finite preorders.