Topology is concerned with those properties that remain invariant under continuous transformations. In the context of Klein’s Erlanger Programm (where it receives a brief mention under its old name of analysis situs) it is the “geometry” of groups of continuous invertible transformations, or homeomorphisms. The “space” to which transformations are applied and indeed the meaning of “continuous” remain somewhat open. When these terms are interpreted in the most general way, as subject only to certain axioms (which we shall not bother to state here), one has general topology. The theorems of general topology, important in fields ranging from set theory to analysis, are not very geometric in flavor. Geometric topology, which we shall be concerned with in this chapter, is obtained when the transformations are ordinary continuous functions on ℝ n or on certain subsets of ℝ n .
KeywordsFundamental Group Euler Characteristic Orientable Surface Nonorientable Surface Face Angle
Unable to display preview. Download preview PDF.