This thesis is logically divided into three chapters. First chapter is generally about computability and it forms a basis for further studies. In the second chapter we talk about computable metric spaces. In order to introduce computability into metric spaces, we define computable points, computable sequences, computable sets, computably enumerable sets, semicomputable sets. We prove that if a semicomputable set S is computable in a point, then we can find a computable neighbourhood of that point in S which has arbitrarily small diameter. As a consequence of that and results from [5], we get that computable points are dense in semicomputable manifolds and semicomputable topological polyhedra. The last chapter is about computable topological spaces. Similarly as in previous chapter, we introduce notions which we use to establish some connections between the computability theory and topology. We investigate sufficient conditions under which semicomputable chainable and circularly chainable continua are computable. We show that if a Hausdorff continuum is semicomputable and chainable from a to b, where a and b are computable points, then it is computable. Moreover, we prove that semicomputability of a Hausdorff continuum which is circularly chainable, but not chainable, implies its computability