By Solomon Lefschetz
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Extra info for Algebraic Topology (Colloquium Pbns. Series, Vol 27)
S) an equia continuous mapping (Set Theory R, § 5, (Set Theory R, § 5, This is a particular case of a general property of quotient structures (Set Theory, Chapter IV, § 2, no. 6, criterion CST 20). PROPOSITION 7 (Transitivity of quotient spaces). Let Rand S be two equivalence relations on a topological space X such that R implies S, and let SIR be the quotient equivalence relation on the quotient space X/R (Set Theory R, § 5, no. 9). Then the canonical biJection (X/R)/(S/R) -- XIS is a homeomorphism.
Closed) in AI' if and Clearly if B is open (resp. closed) in X, then B n AI is open (resp. closed) in A,. Conversely, suppose first that condition a) is satisfied; since (CB) n A, = A, - (B n A,), it is enough, by duality, to consider the case in which each of the B n AI is open with respect to A,. In this case B n A, is open in AI for each tel, and therefore open in X; and since B = U (B n A,) by hypothesis, it follows that B is open in X. I Nowsuppose that b) is satisfied; by duality again, we need only consider the case in which each of the B n A, is closed in AI> and therefore closed in X.
If f: X ~ Y is -1 continuous and if each of the fT( is a homeomorphism of f (T (L) ) onto T (L) , then f is a homeomorphism of X onto Y. t) For f is clearly bijective, and is open by virtue of Proposition 2. 2. OPEN EQUIVALENCE RELATIONS AND CLOSED EQUIVALENCE RELATIONS An equivalence relation R on a topological space X is said to be open (resp. closed) if the canonical mapping of X onto X/R is open (resp. closed). DEFINITION 2. It comes to the same thing to say that the saturation of each open (resp.
Algebraic Topology (Colloquium Pbns. Series, Vol 27) by Solomon Lefschetz