By Aleksandr Sergeevich Mishchenko

This can be primarily a textbook for a latest direction on differential geometry and topology, that is a lot wider than the conventional classes on classical differential geometry, and it covers many branches of arithmetic an information of which has now develop into crucial for a contemporary mathematical schooling. we are hoping reader who has mastered this fabric should be capable of do autonomous learn either in geometry and in different comparable fields. to achieve a deeper realizing of the fabric of this e-book, we advise the reader may still remedy the questions in A.S. Mishchenko, Yu.P. Solovyev, and A.T. Fomenko, difficulties in Differential Geometry and Topology (Mir Publishers, Moscow, 1985) which was once especially compiled to accompany this path.

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6 In Zeeman’s topology for spacetime, every function on the light cone is continuous, as any function will be continuous if the domain space is discrete. With these repeated references to Zeeman’s work we want to stress the following point: people talk frequently of continuous wavefunctions and fields on spacetime but nobody really knows the meaning of that, as the real topology of spacetime is unknown. Curiously enough, much more study has been dedicated to the topological properties of functional spaces (such as the spaces of quantum fields) than to those of the most important space of all Physics.

Any function defined on a space with the discrete topology is automatically continuous. On a space with the indiscrete topology, no function of interest can be continuous. That is what we meant when we said that no useful definition of the continuity concept was possible in these two cases. 6 In Zeeman’s topology for spacetime, every function on the light cone is continuous, as any function will be continuous if the domain space is discrete. With these repeated references to Zeeman’s work we want to stress the following point: people talk frequently of continuous wavefunctions and fields on spacetime but nobody really knows the meaning of that, as the real topology of spacetime is unknown.

But we might be able to say still better, “p is quite distinct from q because p belongs to the neighbourhood U , q belongs to the neighbourhood V , and U and V are disjoint”. To make these ideas precise and operational is an intricate mathematical problem coming under the general name of separability . We shall not discuss the question in any detail, confining ourselves to a strict minimum. The important fact is that separability is not an automatic property of all spaces and the possibility of distinguishing between close points depends on the chosen topology.