توبولوجي عام 3041 uyuioyuiy
كــــــــلــــيــــــة : التربية قـــــــــــســـــــم : قسم الرياضيات
الفصل الدراسي: الــمــســتــــوى : الفرقة الرابعة
كـــــود المقـرر : 09418mat توصيف المقرر :       ▼
المحتوى العلمي:  Topology of the line and plane :- Real line, open sets, accumulation points, Bolzano -Weierstrass theorem. Sequences, convergent sequences. Sub sequences. Cauchy sequences. Completeness. Continuous functions, topology of the plane - Topological spaces: definitions :- Topological spaces, accumulation points. Closed sets. Closure of a set. Interior. Exterior. Boundary. Neighborhoods and neighborhood systems. Convergent sequences. Coarser and finer topologies. Subspaces, relative topologies. Equivalent definitions of topologies - Bases and sub bases :- Base for a topology. Sub bases. Topologies generated by classes of sets. Local bases - Continuity and topological equivalence : Continuous functions. Continuous functions and arbitrary closeness. Continuity at a point. Sequential continuity at a point. Opened and closed functions. Homomorphic spaces. Topological properties. Topologies induced by functions - Separation axioms :- T1-spaces. Hausdorff spaces. Regular spaces. Normal spaces. Urysohn’s lemma and metrization theorem. Functions that separate points. Completely regular spaces - Compactness :- covers, compact sets. Subset of compact spaces. Finite intersection property. Compactness and Hausdorff spaces. Sequentially compact sets. Count ably compact sets. Locally compact spaces. Compactification. Compactness in metric spaces. Totally bounded sets. Lebesgue numbers for covers