We investigate singularities of space-time using the theory of differential spaces in the sense of R. Sikorski [the author and Z. Żekanowski, Demonstr. Math. 20, No. 3/4, 477-487 (1987; Zbl 0665.58003), R. Sikorski, Colloq. Math. 18, 251-272 (1967; Zbl 0162.251) and ibid. 24, 45-79 (1971; Zbl 0226.53004)]. If space-time is modeled by a differential space rather than by a differential manifold, space-time singularities can be regarded as points of the differential space in question. The theory of differential spaces opens some possibilities to classify singularities of space-times [C. J. S. Clarke, Gen. Relativ. Gravitation 6, 35-40 (1975; Zbl 0355.53007) and C. F. R. Ellis and B. G. Schmidt, ibid. 8, 915-953 (1977; Zbl 0434.53048)]. In Section 3 we present such a classification. The differential space methods turn out to be a very efficient tool in dealing with the classical singularity problems [C. F. R. Ellis and B. G. Schmidt, loc. cit. and J. Gruszczak, M. Heller, and P. Multarzyński, J. Math. Phys. 29, 2576-2580 (1988)].
In Section 1 we recall necessary definitions and theorems from the theory of differential spaces. In Section 2 we describe some properties of functions and forms of class \(C^ k\) on a differential space, which are very important in the next sections.