Computably and punctually universal spaces

We prove that the standard computable presentation of the space $C[0,1]$ of continuous real-valued functions on the unit interval is computably and punctually (primitively recursively) universal. From the perspective of modern computability theory, this settles a problem raised by Sierpiński in the 1940s.

We prove that the original Urysohn's construction of the universal separable Polish space $U$ is punctually universal. We also show that effectively compact, punctual Stone spaces are punctually homeomorphically embeddable into Cantor space $2^{\omega }$; note that we do not require effective compactness be primitive recursive. We also prove that effective compactness cannot be dropped from the premises by constructing a counterexample.