Syllabus
1. A concept of a topological space Open and closed sets; interior and closure; neighborhood systems; base of topology, base of neighborhoods of a point; countable weight and countable character, separability; convergence of sequences and nets (* of filters) and Hausdorff spaces (* T_0-, T_1-spaces); continuous mappings; examples of metrizanle and non-metrizable spaces.
2. Operations on topological spaces Subspace, sum and quotient, product; projective (weal, initial) generation, inductive (strong, final) generation; preserving properties; countable product of metrizable (completely metrizable, compact metrizable) spaces, Hulbert cube.
3. Completely regular spaces - embedding into a power of reals Embedding lemma (diagonal product); complete regularity and its preserving by products and subspaces; embedding into Tichonov cube (power of reals); embedding of separable metrizable space into Hilbert cube (* metrizablity of T_3-spaces with countable base).
4. Normal spaces - extension of real-valued functions Normal space and example of metrizable spaces; * counterexamples to preserving by subspaces and products; Urysohn lemma; Urysohn extension theorem; complete regularity of T_4-spaces.
5. Compact and Lindelof spaces Definition by means of covers; characterization by means of nets (* filters, ultrafilters); preserving by continuous images, by closed subspaces; countable and sequential compactness; example of metrizable spaces; extremes and boundedness of real-valued functions; normality of Lindelof spaces; *product of Lindelof spaces that is not Lindelof.
6. Function spaces on compact sets Space C(K); algebras and lattices of continuous functions; Stone-Weierstrass theore; consequences.
7. Tichonov theorem and Cech-Stone compactification, extension of meppings Proof of compactness of products; compactness of Tichonov cube; compactifications; Cech-Stone compactification; extension of continuous mappings; * ultrafilters and beta-hull of N.
8. Cech-completeness and Baire theorem Topological completeness of metrizable spaces; completion of metrizable space; Cech-completeness; examples of locally compact and completely metrizable spaces; Baire theorem; *uniform space and its completion.
9. Topological groups Topological group; uniformities on topological groups; complete regularity.
Annotation
An elementary course in general topology for bachelor's program in General Mathematics.
Recommended for specialization Mathematical Analysis.