In the first two chapters the bare essentials of elementary point set topology are set forth with some hint of the subject's application to functional analysis.Chapters 3 and 4 treat fundamental groups, covering spaces, and simplicial complexes. For this approach the authors are indebted to E. Spanier. After some preliminaries in Chapter 5 concerning the theory of manifolds, the De Rham theorem (Chapter 6) is proven as in H. Whitney's Geometric Integration Theory. In the two final chapters on Riemannian geometry, the authors follow E. Cartan and S. S. Chem. (In order to avoid Lie group theory in the last two chapters, only oriented 2-dimensional manifolds are treated.)

Chapter Some point set topology

　1.1 Naive set theory

　1.2 Topological spaces

　1.3 Connected and compact spaces

　1.4 Continuous functions

　1.5 Product spaces

　1.6 The Tychonoff theorem

Chapter 2 More point set topology

　2.1 Separation axioms

　2.2 Separation by continuous functions

　2.3 More separability

　2.4 Complete metric spaces

　2.5 Applications

Chapter 3 Fundamental group and covering spaces

　3.1 Homotopy

　3.2 Fundamental group

　3.3 Covering spaces

Chapter 4 Simplicial complexes

　4.1 Geometry of simplicial complexes

　4.2 Baryccntric subdivisions

　4.3 Simplicial approximation theorem

　4.4 Fundamental group of a simplicial complex

Chapter 5 Manifolds

　5.1 Differentiable manifolds

　5.2 Differential forms

　5.3 Miscellaneous facts

Chapter 6 Homology theory and the De Rham theory

　6.1 Simplicial homology

　6.2 Do Rham's theorem

Chapter 7 Intrinsic Riemannian geometry of surfaces

　7.1 Parallel translation and connections

　7.2 Structural equations and curvature

　7.3 Interpretation of curvature

　7.4 Geodesic coordinate systems

　7.5 Isometrics and spaces of constant curvature

Chapter 8 Imbedded manifolds in Ra

Bibliography

Index