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1. Let X be given the co-finite topology. Suppose that X is a infinite set. Is X Hausdorff? Should compact subsets of X be closed? What are compact subsets of X?
2. Let (X,T) be a co-countable topological space. Show that X is connected if it is uncountable. In fact, show that every uncountable subspace of X is connected.
3. Let X be a set and F is separating collection of functions f: X --> Y_f, each from X into a topological space Y_f. Prove that X with the weak topology by F is Hausdorff.
4. Explain how we can think of the unit sphere (a subspace of IR^3 with its usual topology) as a subset of the Hilbert cube.
What similarities and differences do you see between functions and linear equations that you studied in Chapter 3? Are all linear equations functions?
Compute the integral of xdz (|z|=r) for the positive sense of the circle in two ways first by using parametrization and second by observing that x=(1/2)(z+z conjugate)=(1/2)(z+r^2/z) on the circle.
Describe how a domestic fast food chain like McDonald's with plans for expanding into China would be able to use a forecasting model.
A tree has 11 vertices of degree 3, 12 vertices of degree 2, 10 vertices of degree 4 and the remaining vertices are of degree 1. How many vertices does it have?
What is the probability of rolling a total of 7 in two rolls of a standard dice if you get an even number on the first roll?
Analyze the stability of the fixed points of linear differential systems. Provide several explicit examples, with graphical illustrations
Explain Description of Isomorphism, We are working on the proof of showing G (the group of rigid motions of a regular dodecahedron) is isomorphic to the alternating group A_5.
Determine the number of groups under a given condition and How many license plates can a state have if each license plate has 2 letters followed by 4 digits, if the first digit cannot be zero?
Describe your question and receive a step-by-step response, usually within an hour or two.Find the volume of the solid bounded by x^2/(16 ) + y^2/9 = 1 revolving about the x-axis:
Explain the concept of inverse variation. Provide two examples. What is the vertical asymptote for the function y = ( x^2 - 4) / ( x - 2 )?
Repeat the procedure making a second graph for the third-order rational function (e.g., x3). For each of the two graphs, describe how changes in coefficients and constants change the behavior of the function.
The probability that both defective lights will be found. A box contains two defective Christmas tree lights that have been inadvertently mixed with eight non-defective lights. If the lights are selected one at a time without replacement and teste..
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