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Geometry and Topology

Topology : Subspace

Suppose (X,T) is a topological space. Let Y be non-empty subset of X. The the set J={intersection(Y,U) : U is in T} is called the subspace toplogy on Y. Prove that J indeed a toplogy on Y i.e., (Y,J) is a topological space.

Topological Space : Subspace

19. Let X be a topological space and let Y be a subset of X. Check that the so-called subspace topology is indeed a topology of Y. (question is also included in attachment)


14. Make a Mà¶bius strip out of a rectangle of paper and cut it along its central circle. What is the result? 15. Cut a Mà¶bius strip along the circle which lies halfway between the boundary of the strip and the central circle. Do the same for the circle which lies one-third of the way in from the boundary. What are the r

Undergrad 400 level Topology.

Find a tree in the polyhedron of figure 1.3 which contains all the vertices. Construct the dual graph Г and show that Г contains loops. (You don't have to construct the graph, but please describe it to me how it looks like.) (SEE ATTACHMENT)

Undergrad Topology - 400 Level

1. Prove that v(Г) - e(Г) = 1 for any tree T. (v :vertices and e : edges) 2. Even better, show that v(Г) - e(Г) ≤ 1 for any graph Г, with equality precisely when Г is a tree.

Topology : Connected Spaces and Explosion Point

Please see the attached file for the fully formatted problems. B6. (a) Define what it means for a topological space to be connected. (b) Suppose that A and B are subspaces of a topological space X, and that U C A fl B is open in both A and B in the relative topologies. Show that U is open in A U B in the relative topology.

Fubini Type I : Volume

Find the volume of the region in R3 bounded by z = 1 - x2, z = x2-1, y + z =1 and y= 0.

Minimize Cost for a Cylindrical Can

The metal used to make the top and bottom of a cylindrical can costs 4 cents/in^2, while the metal used for the sides costs 2 cents/in^2. The volume of the can is to be exactly 100 in^3. What should the dimensions of the can be to minimize the cost of making it? Could you please show all work so I can better grasp the conce

Maltitudes, Circumcircles and Circumcenters

For any quadrilateral one can define the so-called maltitudes. A maltitude on a side of a quadrilateral is defined as the line through the midpoint of the side and perpendicular to the opposite side. Generally the four maltitudes of a quadrilateral are not concurrent, but if the quadrilateral is cyclic they are. Prove that i

Compact and perfect sets

If P is a perfect set and K is compact is the intersection P intersection K always compact?always perfect?.


Let S = {(x,y,z):2x-y+3z=6},p=(1,0,-1) belonging to R3, and let f:S->R be defined by f(q)=d(q,p), ie, f(x,y,z)=sqrt((x-1)^2+y^2+(z+1)^2) Question: Is S compact? Please verify if q belongs to S and q does NOT belong to [-5,5]^3, then f(q) >= 4. ALSO prove that f attains its absolute minimum value at some point q0 belongi


If V is the volume of a cube with edge length x and the cube expands as time passes, find dV/dt in terms of dx/dt.

Find all possible lengths, widths & heights of a given volume of a prism

I need to know how to find all possible lengths, widths & heights of a given volume of a rectangular prism. I'm writing a program in Java that takes the user inputted volume of a rectangular prism and then tells the user all of the possible lengths, widths & heights are for that given volume. I just don't know the calculations t


If 2400 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Volume = cubic centimeters.

Dimensions of a Rectangle Given the Area

James wanted a photo frame 3 in. longer than it was wide. The frame he chose extended 1.5 in. beyond the picture on each side. Find the outside dimensions of the frame if the area of the unframed picture is 70 in.^2 (square inch.)

Minimization of area

A piece of wire 35cm long is to be cut into two pieces,one piece will be bent into a equilateral triangle and the other a circle.How should the wire be cut so that the enclosed area is minimized?

Cylindrical area and volume

Solve the following word problem, you will need to know that a cylinder of radius r and height h has volume V = Pi r^2 h. Also, a circle of radius r has area A = Pi r^2 and circumference C = 2 Pi r. A closed cylindrical can has a radius t and height h. a) If the surface area S of the can is a constant, express the volume

Complex Variable (Circle; Contour)

See attachment for better formula representation and theorem 5. For each of the following use Theorem 5. to establish the indicated estimate. a. If C is the circle |z| = 3 traversed once then: | Integral C ( dz/ z^2 - i) | < 3pi / 4

Angles and Degrees

Determine the angle when (6 angle (120deg))*(-4+jb+2*exp(j*15))=a+jb Note: These angles are in degrees. Possible choices: a. -21.3 b. -12.1 c. -3 d. -2.07

Create constructions-

Construction- To construct two angles the same measurement Please construct the following. Please make it large enough. not very small thank you Step1. Draw an acute angle. Label the vertex P. Step 2. Use a straightedge to draw a ray on your paper. Label the endpoint T. Step 3. With P as the center , draw a large arc

Finance : Dimensions of a Container

A beverage is sold in bottles that are 7 inches high and 2 inches at the base. The bottles are packed 12 to a case (2 rows and 6 bottles to a row). The case is made of 1/8 inch thick corregated cardboard with 1/16-inch cardboard dividers. An extra layer of 1/8-inch cardboard is placed on the bottom for a cushion. What are the di

Finance : Dimensions of a Container

You work in a goblet factory.A client has ordered 240 goblets. You must pack each goblet in individual boxes before placing them in containers for shipmnt. The measurements foe each box is 3inx6inx6in. The shipping container is made of cardboard 1/8-inch thick.No spacers are needed. What are the dimensions of the containers if

Geometry Transformations Questions

1) A double reflection over two intersecting lines is the same as a single_____. 2)A double reflection over two parallel lines is the same as a single ______. Given the ordered pair (-3,5), give the coordinates of its image after each of the following tranformations: 1) A reflection over the line y=x _______ 2) A ref

Countable and normal

A) Reals with the "usual topology." Is there a way to prove this space is normal other than just saying it is normal because every metric space is normal? b) Reals with the "K-topology:" basis consists of open intervals (a,b)and sets of form (a,b) - K where K = {1, 1/2, 1/3, ... } Why connected? Why 2nd countable?

Volume of a Sphere : Normalized Volume of Spherical Band

Please see the attached file for the fully formatted problems. Consider the sphere x20 + x21 + · · · + x2n = n of radius sqrt(n). Show that the normalized volume of the spherical band where a <= x0 <= b is .... Hint: 1 &#8722; x =< e^&#8722;x will be helpful at one point.

Working with Topological Spaces

Which of the following topological spaces is normal? a) Reals with the "usual topology." b) Reals with the "finite complement topology:" U open in X if U - X is finite or is all of X. c) Reals with the "countable complement topology:" U open in X if X - U is countable or is all of X. d) Reals with the "lower limit topolog

Volume of Solid of Revolution

Please see the attached file for the fully formatted problems. 1.) Sketch 2.) Show a typical slice properly labeled 3) Write the formula for the volume of the shell generated 4) Set up the corresponding integral 5) Evaluate the integral Y = x^2 , y = 3x, about the y-axis.