Math 440, Fall 2014, Assignment 5
From cartan.math.umb.edu
I was at the mathematical school, where the master taught his pupils after a method scarce imaginable to us in Europe. The proposition and demonstration were fairly written on a thin wafer, with ink composed of a cephalic tincture. This the student was to swallow upon a fasting stomach, and for three days following eat nothing but bread and water. As the wafer digested the tincture mounted to the brain, bearing the proposition along with it.
- - Jonathan Swift, Gulliver's Travels
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Weak topology (induced by a collection of maps).
- Product (of two topological spaces).
- Product (of an arbitrary collection of topological spaces).
- Strong topology (induced by a collection of maps).
- Quotient space (of a topological space by an equivalence relation).
- Coproduct (or "disjoint union") of two topological spaces.
- Coproduct (of an arbitrary collection of topological spaces).
Carefully state the following theorems (you need not prove them):[edit]
- Theorem characterizing continuity of maps into weak topologies (in terms of certain compositions).
- Theorem characterizing continuity of maps from strong topologies (in terms of certain compositions).
Solve the following problems:[edit]
- Problems 8D(1-3), 9A(1), and 9A(3) (the last part, concerning $S^2$, is quite difficult at this stage, though it will become easier towards the end of the course; consider it optional).
- The box topology. Let $\{X_{\lambda}\}_{\lambda\in\Lambda}$ be any collection of topological spaces. Define a new topology $\tau$ on $\prod_{\lambda\in\Lambda}X_{\lambda}$ (called the box topology) by taking as a base the collection of all sets of the form $\prod_{\lambda\in\Lambda}U_{\lambda}$ where each $U_{\lambda}$ is open in $X_{\lambda}$ (these sets are called open boxes). Show that, whenever $\Lambda$ is finite, the box topology coincides with the ordinary product topology. Give examples to show that, when $\Lambda$ is infinite, the box topology may or may not equal the product topology. In the infinite case, which of these topologies do you think is more frequently used? Why?
