Some sets of first category (Q1319412)

In what follows similarity is an order isomorphism. The first part of the paper contains some interesting observations concerning sets of first category whose images under similarities are of first category. Sample result: If \(A\subseteq \mathbb{R}\) is dense in some open interval \(I\) and \(A\cap I\) is a set of first category, then every set similar to \(A\cap I\) is of first category. Further, let \({\mathcal P}_ 1= \{A\subseteq \mathbb{R}\): \(A\) is similar to a dense set of first category\}. Sample result: Let \(A\in {\mathcal P_ 1}\) and let \(f: A\to f(A)\subseteq \mathbb{R}\) be a homeomorphism, then \(f(A)\) is a set of first category. An example is provided showing that the above fails if \(f\) is assumed to be continuous. J. C. Morgan II established the existence of a linear set every homeomorphic image is of first category, but which is not always of first category. In view of the previously mentioned result and that \(\{A\in {\mathcal P}_ 1\): \(A\) is not a set possessing the property always of first category\(\}\neq \emptyset\) (actually of cardinality \(\geq 2^{\aleph_ 0}\)), the author finds that there are ``many'' such sets possessing the property described by Morgan.