I am unclear on the concept of dimensionality when it comes to using Monte Carlo for option valuation and what I am reading has unfortunately not made things any clearer.

Starting as simply as possible, simulating the terminal price at option maturity of a single underlying in one time step jump from valuation to maturity is presumably an example of a one dimensional problem. Is doing the same thing for multiple underlyings (e.g. a European basket option) a multi-dimensional problem?

What is the dimensionality if I value a single underlying vanilla option by evolving the underlying price over a path using multiple time steps DT = T/N, where T is the option maturity in years and N is the number of time steps. Is that a multi-dimension problem? If so, why is this so?