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توبولوجي عام 3041
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uyuioyuiy
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كــــــــلــــيــــــة :
التربية
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قـــــــــــســـــــم :
قسم الرياضيات
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الفصل الدراسي:
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الــمــســتــــوى :
الفرقة الرابعة
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كـــــود المقـرر :
09418mat
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توصيف المقرر :
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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
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