Syllabus For The Subject SET TOPOLOGY AND THEORY OF RELATIVITY

UNIT - 01

TOPOLOGICAL SPACE - DEFINITION - DIRECT TOPOLOGY - FINER THEN BASIS FOR A

TOPOLOGY - STANDARD TOPOLOGY - SUB BASIS TOPOLOGY - ORDER TOPOLOGY -

RAYS - PRODUCT OF TWO TOPOLOGY - SUBSPACE TOPOLOGY - IF A IS SUBSPACE OF X

AND B IS SUBSPACE OF Y THEN PRODUCT OF TWO TOPOLOGY.

UNIT - 02

A X B IS SAME - CONVEX - LET X BE AN IN THE ORDERED TOPOLOGY LET Y BE A SET OF

X THAT IS CONVEX IN X THEN THE ORDERED TOPOLOGY ON Y IS THE SAME AS THE

TOPOLOGY ON Y IS THE SAME.

UNIT - 03

AS THE TOPOLOGY Y-CLOSED SETS - LIMIT POINTS - LET Y BE A SUBSPACE OF X THEN

A SET A IS CLOSED IN Y IF IT EQUALS THE INTERSECTION OF CLOSED SETS OF X WITH

Y.

UNIT - 04

CLOSURE SETS - INTERIOR SETS - LIMIT POINTS - HAUSDORFF - EVERY FINITE POINT IN

A SET IN A HAUSDORFF SPACE IS CLOSED - IF X IS HAUSDORFF SPACE THEN

SEWUENCE OF POINTS OF X CONVERGES TO AT MOST ONE POINTS.

UNIT - 05

A SUBSPACE OF HAUSDORFF SPACE IS HAUSDORFF - CONTINUES FUNCTION -

HOMOMRPHISM - UNIT CIRCLE - CONSECRATING CONTINUES FUNCTION.

UNIT - 06

PASTING LEMMA - MAPS INTO PRODUCTS - PRODUCT TOPOLOGY - CARTESIAN

PRODUCT - BOX TOPOLOGY -COMPARISON OF BOX AND PRODUCT TOPOLOGY.

UNIT - 07

METRIC TOPOLOGY - DISTANCE - BALL - METRIZABLE - BOUNDED DIAMETER -

SLANDERED BOUNDED METRIC - EUCLIDEAN - UNIFORMETRIC.

UNIT - 08

SEQUENCE LEMMA - FIRST COUNTABLE AXIOM - SECOND COUNTABLE AXIOM -

UNIFORM LIMIT THEOREM - QUOTIENT TOPOLOGY - CONNECTED SPACE - COLLECTION

OF CONNECTED SUBSPACE COMMON POINT.

UNIT - 09

THE IMAGE OF CONNECTED SPACE CONTINUOUS MAPPING - A FINITE CARTESIAN

PRODUCT OF CONNECTED SPACE - SUBSPACE OF REAL LINE - INTERMEDIATE VALUE

THEOREM.

UNIT - 10

COMPACT SPACE - COVER - OPEN COVER - COMPACT - EVERY CLOSED SUBSPACE OF A

COMPACT SPACE IS COMPACT - EVERY COMPACT SUBSPACE OF A HAUSDORFF OF

SPACE IS CLOSED - THE IMAGE OF COMPACT SPACE UNDER A CONTINUES MAP IS

Vinayaka Missions University,Directorate of Distance Education

Salem India

MASTER OF SCIENCE IN MATHEMATICS

1 Yr.

SET TOPOLOGY AND THEORY OF RELATIVITY{MMAT.05}(2030508)

COMPACT - A PRODUCT OF FINITELY MINING COMPACT SPACE IS COMPACT - TUBE

LEMMA - FINITE INTERSECTION PROPERTY - COMPACT SUBSPACE OF REAL LINE

EXTREME VALUE THEOREM - LEBESGUE NUMBER LEMMA.

UNIT - 11

UNIFORM CONTINUITY THEOREM - CONTINUES COMPACT REAL LINE - COUNT ABILITY

ACTION - FIRST COUNTABLE - SECOND - DENSE - LINDALE SPACE - SEPARABLE -

SEPARATION ACTIONS - REGULAR - NORMAL - A PRODUCT OF HAUSDORFF SPACE IS

HAUSDORFF - SUBSPACE OF HAUSDORFF SPACE - A SUB APACE OF REGULAR SPACE

IS REGULAR - A PRODUCT OF REGULAR SPACES REGULAR - EVERY REGULAR SPACE

WITH A COUNTABLE BASIS IS NORMAL - EVERY COMPACT- HAUSDORFF SPACE IS

NORMAL - EVERY WELL ORDERED SET X IS NORMAL IF THE ORDERED TOPOLOGY -

URYSOHN LEMMA - COMPLETELY REGULAR -A SUBSPACE OF COMPLETELY REGULAR

SPACE COMPLETERLY REGULAR - A PRODUCT OF COMPLETELY REGULAR SPACE IS

COMPLETELY REGULAR.

UNIT - 12

URYSOHN METRIZATION THEOREM - IMBEDDING THEOREM - THE TIETZE EXTENSION

THEOREM - MINIMAL UNCOUNTABLE WELL ORDERED SET - COMPONENTS AND LOCAL

CONNECTED.