How Peano's space-filling curves sparked a century of deep research in topology, with more to come

Giuseppe Peano's sensational 1890 discovery that the closed unit interval [0, 1] can be continuously mapped onto its square and cube naturally gave rise to the problem of what spaces can be its continuous images. This was settled by the beautiful Hahn-Mazurkiewicz theorem (1914, 1920) that said these images are precisely the compact, connected, locally connected, metrizable spaces. Topologists also characterized [0, 1] topologically as the only linearly orderable, compact, connected, metrizable space with more than one point.

A study naturally began of spaces with the same description but with "metrizable" omitted, giving rise also to the question of characterizing the continuous images of these spaces. A number of complicated characterizations were arrived at in the latter half of the 20th century, but it was only near the end that the exceptionally deep researches of Treybig, Nikiel, and Mary Ellen Rudin culminated in the beautiful characterization that puts "monotonically normal" for "metrizable" in the Hahn-Mazurkiewicz theorem.

This seems to be the end of this story if we only use the usual (ZFC) axioms. But if we assume some axioms that negate the familiar Continuum Hypothesis (CH) it is still unknown whether "monotonically normal" can be replaced with the more elementary property that every subspace is normal. Some research in this direction will be summarized.