A pioneering result in logic is Cantor's theorem on the \(\omega\)-categoricity of DLO. One way of phrasing this result is to say that every pair of countable dense sets of reals \(A, B \subseteq\mathbb R\) are linear order isomorphic. A moment's reflection confirms that this in turn implies that the real line is a CDH topological space: for every pair of countable, dense sets \(A, B \subseteq \mathbb R\) there is an autohomeomorphism \(h:\mathbb R \to \mathbb R\) mapping \(A\) to \(B\). Brouwer later showed that the same holds in the higher dimensional finite Euclidean spaces \(\mathbb R^n\).

From the perspective of set theory of the reals it is natural to ask whether these results remain valid when "countable" is exchanged with "uncountable". The Baumgartner axiom, hereafter BA, states exactly this for the one dimensional case: every pair of \(\aleph_1\)-dense sets of reals are order isomorphic. Here \(\aleph_1\)-dense means that the intersection with every open set has size \(\aleph_1.\) It's clear that BA implies the failure of the continuum hypothesis but Baumgartner showed that BA was consistent with the axioms of set theory in 1973 using a still notoriously tricky forcing argument. Later, in 1989 Steprāns and Watson showed the analogue of BA can hold consistently at the higher dimensional \(\mathbb R^n\)'s as well as finite dimensional, compact manifolds of dimension at least 2. Somewhat mysteriously the higher dimensional versions are not equivalent to the one dimensional version which seems "harder" in some difficult to quantify way. This led to the Steprāns-Watson conjecture: BA implies its higher dimensional analogues. In this talk we will sketch this landscape focussing on recent work of the speaker to introduce intermediate BA-like axioms which clarify the possible applications of these ideas in topology and analysis.