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The intersection of two open sets is compact iff it is empty. Can the intersection of an infinite collection of open sets be a non-empty compact set?
Prove that X is compact if and only if for each family {F_a} with a in I of closed subsets of X that has the finite intersection property, the intersection.
Show that each subset of S is compact and that therefore there are compact subsets of S that are not closed.
Show that, if Y has the discrete topology and if p: X x Y --> X is the projection onto the first factor, then p is a covering map.
Define a new metric d on X = (0, 1/2)2 by d((a,b), (r,s)) = 1 if a is not equal to r Or |b - s| if a = r.
Find the interior, the closure, the accumulation points, the isolated point and the boundary points of the following sets.
A particle of mass 2 moves along the x-axis and is attracted towards the origin O by a force F=-8xi.
Suppose a twisted curve is defined in terms of the arclength s by where ? is a constant parameter.
Use the shell method to find the volume of the solid generated by revolving the regions bounded by the curves and lines about the y-axis.
Suppose that a small brick is put into an aquarium which is the shape of a rectangular solid 1 foot wide, 3 feet long, and 2 feet high.
A gallon of paint covers about 350 square feet. How many gallons would be required to paint the room? Round up to the nearest gallon.
Water is poured into a funnel at the constant rate of 1 in^3/sec and flows out at a rate of 1/2 in^3/sec.
Systems of equations can be solved by graphing, using substitution, or elimination. What are the pros and cons of each method?
Use the QR factors in 1) to determine the lease squares solution to Ax=b. Ax=b is the solution related to Rx=Q^Tb
Determine whether it is possible to find a circumscribable, isosceles "true" trapezoid (in which the parallel sides are not congruent).
It is a common experience to hear the sound of a low flying airplane, and look at the wrong place in the sky to see the plane.
Find the volume of the pyramid with a rectangular base 11 in height, 12 in length, and 10 in width.
An ellipse with vertices at (-2,5) and (-2,1) and a minor axis 2 units long. A circle where (4,-1) and (-6,6) are endpoints of a diameter.
What does it mean to refer to a 20 inch TV set or a 25 inch TV set? Such units refer to the diagonal of the screen.
A = { A a subset of X : X - A is either finite or countable} U { X,0} is a topology on X ?. Why ? ( note: the 0 means the empty set).
Points A, B,C and D are on a square. The area of teh squre is 36square units. Which is true about the perimeter of rectangle ABCD?
Find the exact value of the volume of a can of asparagus with a diameter of 8 centimeters and height of 10 centimeters.
Find the approximate value of the volume of the right circular cone with a circular base shown below. Approximate your solution to the nearest hundredth.
Find a nonzero vector x_4 such that {x_1, x_2, x_3, x_4} is a set of mutually orthogonal vectors.
Apply the Gram-Schmidt process to the vectors above in reverse order: in the form A = QR. where a_1= (1 1 1)^T, a_2=(0 1 1)^T, and a_3=(0 0 1)^T