*Homework* 4 *Solutions* Let X = producttext X i , and let i : X X i be the projection on the i-th factor. To see how evil alternatives to the product *topology* are, consider X = S 1 S 1 with the box *topology*, where for definiteness were taking S 1 to be the unit circle in the complex plane. A path is continuous if all components i are continuous and all but finitely many of the i s are constant . *Homework* 4 *Solutions*. Problem 13.4. Proposition. If {Tα} is a family of topologies on X, then Tα is a *topology* on X. Proof.

*Topology* *homework* *solution* Note that a map : [0 , 1] producttext X i is continuous if and only if each map i = i : [0 , 1] X i is continuous. In other words, a homotopy class of loops in X is a seqence of homotopy classes of loops in X i , or 1 ( X, x ) = producttext 1 ( X i , x i ), where x = ( x 1 , x 2 , . (In the box **topology**, the set U 1 U 2 is open if each U i is open. To see this, suppose that infinitely many i s are non-constant, and WLOG we can assume that 1 , 2 , . *Topology* *homework* *solution* - ohio dissertation thesis. S17 - *Solutions* to *Topology* *Homework* #4, due Week 8 Math 441 Autumn 2007 *Solutions* to *Homework* 5.

The Clash Over the Poincaré Conjecture - The New Yorker **Topology** is all about the study of properties in mathematics, preserved in the forms of twistings, stretchings of various objects and deformations. By the nineteen-sixties, *topology* had become one of the most productive. “We felt that nobody else would be able to discover the *solution*,” Yau told us.