Math 429, Quiz 5.
1. Let A be a connected subspace of space X and A Y A. Prove that Y is
2. Let X be a topological space with two points a and b and only three open
sets , cfw_a, and cfw_a, b. Show that X path connected.
Math 429, Quiz 6.
1. Prove that if X is connected and has the same homotopy type as Y then Y
is also cconnected.
2. For a path f let f be the path given by
f (t) = f (1 t).
Prove that f
g (relcfw_0, 1) if and only if f
g (relcfw_0, 1).
Math 429, Quiz 4.
1. Dene on R by x y if and only if x y is rational. Show that is an
equivalence relation and that R/ with the quotient topology is not Hausdor.
2. Let X be a Hausdor space and A, B X be two compact subsets. Show
that A B is co
Math 429, Quiz 3.
1. Suppose that X is a G-space. Prove that the function g given by g (x) = g x
is a homeomorphism from X to itself for all g G.
2. Consider R with equivalence relation x x. Show that R/ is homeomorphic to [0, ).
Math 429, Quiz 2.
1. Show that subset (a, b) in R with induved topology is homeomorphic to R.
2. Give an example of topological spaces X , Y with a continuus bijection
f : X Y such that f 1 is not continuous.
Math 429, Midterm 3.
1. Prove that any continuous map f : D2 D2 has a xed point; i.e., a point
such that f (x) = x.
2. Let p : X X be a covering and let f , f : Y X be two liftings of
f : Y X.
Assume Y is connected and f (y0 ) = f (y0 ) for som
Math 429, Midterm 2.
1. (8.7) Prove that compact subset of Hausdor space is closed.
2. (9.3) Show that interval [0, 1] is connected.
3. Let be a relation on the points of a space X dened by saying that x y
if and only if there is a path joining
Total internal reflection ellipsometry: principles
Hans Arwin, Michal Poksinski, and Knut Johansen
A concept for a measurement technique based on ellipsometry in conditions of total internal reflection is
presented. When combined with sur