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Topology was invented by Henri Poincaré (1850 to 1912) who called it *analysis situs* to handle stability problems in celestial mechanics. He followed, combinatorial methods devised by Leonard Euler (1707 to 1783) who traced the subject back to Leibniz (1646 to 1716) as well as work in differential geometry by Gauss (1777 to 1855) and Riemann (1826 to 1866).

Topology has developed in several different directions:

- Differential topology: closest to the original Poincaré concept. It studies surfaces, solutions of differential equations, etc.

- Algebraic topology: the study of algebraic (= groups, rings, etc.) invariants of topological spaces.

This led to important developments in algebra, to the development of category theory, etc.

- Combinatorial or geometric topology: developed from early attempts to answer questions in 1) and 2) above.

- General or point-set topology: the basic theory underlying the above. This last is what this course will consider.

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