If two separate topologically inequivalent 1-manifolds (for example 2 curves) were entertained as ‘apposed’ geometrically, or via ‘limit’ i.e. summation series, or 2 tangents’ respective elements (neighborhood of 3 elements?) of respective sets intersecting, to describe degree of closeness (‘limit’ of intersection; a valid construct? see MSM page in https://sites.google.com/site/zankaon); then has topology changed, and does one then have a mixture of continua i.e. co-mingling of elements of 2 open sets? For such mixture of elements, one could no longer utilize a matching of such respective elements, in order to ascertain topological equivalence of such manifolds. Again, indirectly does cardinality of manifolds set constraints on topology? One can seemingly utilize element of a set, or neighborhood of such element; hence immediately referring to a larger context, and thus avoiding the ‘point’ construct.
If manifolds are invariant, could one then focus more on relatedness (i.e. product space in general sense) of such manifolds, even with different topologies? For example, for dimensionality considered as a product space, could one relate a 1-manifold to another 1-manifold, but with a different topology, giving a third manifold of higher dimension (2-manifold), but of mixed topology? And for n-manifold, then a mixture of n-continua?
For another example, might one have relatedness of manifolds of different topologies (mapping to rationals and integers, respectively), giving rise to another manifold, a mixed 4-manifold M3m x M1m -> 4-manifold i.e. modified curvilinear cylinder, somewhat like Fig. 23, Section 68 of SRM monograph?
The coarser to finer 3-manifold is not smooth and hence not analytical, since only finite order of differentiation down to Planck scale. Yet still the continuum is defined, since such coarser to finer scale maps to rationals; hence still 3-mainifold description for less than Planck scale. This is all consistent with a mixture of continua i.e. manifolds; since in contrast, MTC(s) i.e. describing evolving set of entangled manifolds, is always exponentially changing, and hence analytical. Thus the non-uniform flow of cosmic time is always smooth. So the mixture of the two manifolds differs in continuity and smoothness, but not in cardinality.
Our atmosphere appears as a whole, yet it is a mixture of gases. Analogously, we perceive our manifold, and how it evolves, as a whole; but is it actually a mixture of continua? So another road to cosmic time evolution description of manifold(s)? Other roads being MGS construct (not a physical manifold), and also intersecting (emphasizing cosmic time evolution intrinsic to manifold and System) manifolds M3m and M1m. Are then mixtures of continua not that strange? TMM