Components and Equivalence Classes in Mathematical Spaces
Explore the concept of components in mathematical spaces, defined by equivalence relations on a space where each connected subspace contains both x and y. The components are mutually disjoint unions in x, forming a crucial aspect of mathematical analysis.
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Components Dr.M.Gilbert Rani Department of mathematics Arul Anandar College (Autonomous) Karumathur
Definition :- X arbitrary space An equivalence relation on x by setting x y , if there is a connected subspace of X containing both x & y. The equivalence classes are called components of X in other words .. * mutually disjoint * union in x
Theorem 25.1 The components of X are connected disjoint subspaces of X whose union is X, such that each non-empty connected subspace of x intersects only one of them. Proof since the components are by definition of equivalence classes, then the components are disjoint and union in X To prove Each connected subspace A of X eliminates only one of them
Suppose the connected subspace A of X intersects the components C1 and C2 say x1 & x2respectively then x1 x2 X since x1, x2 c1 X1 A, C1 X2 A, C2 A c2 c1 x1, x2 c2 X1 X2 then c1= c2 x1 , x2 A C1, C2 A which is a contradiction Next to prove:- Component C is connected choose a point x0 C , x C then we have x0 x there is a connected space Ax containing x0 and x
since each connected subspace of x intersects only one component Ax C C = x c Ax since each Ax is connected and have a point x0 is common , then x cAx is connected . W.K.T. connected (Theorem- 23.3) The union of collection of connected subspaces of x that have a point in common is then COMPONENT C IS CONNECTED
