I know that for every $k$-tape DTM that runs in time $O(t(n))$, there exists a 1-tape DTM that runs in $O(t^2(n))$, no matter how large the $k$ (the $k$-part is a formulation from Wikipedia). But what if $k=\infty$?
If I understand it right, if $k=\infty$, then the alphabet of the $\infty$-tape DTM is infinite and therefore the transition function will be infinite as well. That is why it is not NTM by definition, however it should be much more powerful than a DTM (based on the argument that we can encode a countable infinitely long input into one symbol of countable infinitely large alphabet).
So, is it known if a NTM can be simulated by $\infty$-tape DTM (with preserving polynomially-same accepting times)?