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 :
T1spaces. 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

