The (sub)copulas play a central role in statistics for capturing the stochastic dependence structure of random variables. A subcopula is a map from the Cartesian product of two closed subsets of the unit closed interval containing zero and one to the unit closed interval, which is grounded, 2-increasing and has uniform margins. A copula is a subcopula whose domain is the Cartesian product of the unit closed interval and itself. The uniform metric on the set of all copulas can be extended to a metric on the set of all subcopulas. In this work, we investigate topological structures of these spaces by using tools from infinite-dimensional topology. Our main result is as follows: there exists a homeomorphism from the space of subcopulas onto the Hilbert cube which sends the space of all copulas onto an end of the Hilbert cube and sends the space of exchangeable copulas onto an end of the end of the Hilbert cube. As a consequence, we obtain that the spaces of subcopulas, copulas and exchangeable copulas are homeomorphic to the Hilbert cube and hence have the fixed point property.