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# Geometry and Topology

### Topologies : Open Sets

? Let X:={a,b,c} be a set of three elements. A certain topology of X contains (among others) the sets {a}, {b}, and {c}. List all open sets in the topology T. ? Let X':={a,b,c,d,e} be a set of five elements. A certain topology T' on X' contains (among others) the sets {a,b,c}, {c,d} and {e}. List any other open set in T' which

### Volume of Hyperspheres

Finding formulas for the volume enclosed by a hypersphere in n-dimensional space. d) Use an n-tuple integral to find the volume enclosed by a hypersphere of radius r in n-dimensional space R^n. (Hint: The formulas are different for n even and n odd.)

### 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

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 on Y.

### 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)

### Topology: Homomorphisms

Define f: [0, 1) →C by f (x) = e2πix. Prove that f is one-one, onto, and continuous. Find a point x ∈ [0, 1) and a neighborhood N of x in [0, 1) such that f (N) is not a neighborhood of f (x) in C. Deduce that f is not a homomorphism. See the attached file.

### 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.

### Understanding Wave Diagrams: Golden Ratio

Please help in understanding a "wave diagram". "The Power of Limits" by Gyorgy Doczi, is about the relationship of shape, music, nature etc to the golden section. A lot of the book has diagrams relating to an object showing how the forms relate to the golden section. One such diagram is a wave diagram, and connects a dinergic di

### 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

### Minimizing the Surface Area of a Box

A closed rectangular box with volume 576 in^3 is to be made so its top (and bottom) is a rectangle whose length is twice its width. Find the dimensions of the box that will minimize its surface area. Could you please show all work so I can better grasp the concept? Thank you.

### 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

### Extreme Value Theorem

(Extreme Value Theorem) prove if f:K->R is continuous on a compact set K subset or equal to R, then f attains a maximum and minimum value.In other words there exists Xo,X1 belong to K such that f(Xo)<=f(X)<=f(X1) for all X belong to K.

### Convex Hull : Closed and Compact Sets

Could you please prove or disprove both: (a) The convex hull of a closed set is closed (b) The convex hull of a compact set is compact.

### Compact and perfect sets

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

### Orthogonal Subspaces

Let A be an mxn matrix. show that 1) If x &#1028; N(A^TA), then Ax is in both R(A) and N(A^T). 2) N(A^TA) = N(A.) 3) A and A^TA have the same rank. 4) If A has linearly independent columns, then A^TA is nonsingular. Let A be an mxn matrix, B an nxr matrix, and C=AB. Show that: 1) N(B) is a subspace of N(

### Dv/dt

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

### Volume

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.)

### Geometric Application Problem

A rectangular building whose depth is twice its frontage is divided into two parts, a front portion and a rear portion, by a partition that is 30 feet from and parallel to the front wall. Identify the front (width) by the letter "x" and write the following: Depth (length) of the building Length of the rear portion Writ

### Constructing two equal angles

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

### 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?

### 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

### Plane Separation : Proving Convexity

If a set of points P1 and P2 are convex subsets of a metric geometry, prove that P1 union P2 is convex.

### Betweenness and Elementary Figures : Taxicab Plane

If A=(4,7), B=(1,1), and C = (2,3) prove that A-C-B in the Taxicab Plane. Please help me to understand.

### Find the Volume of a Sphere with a Cylindrical Hole

#26 Please see the attached file for full problem description. (a) A cylindrical drill with radius r1 is used to bore a hole through the center of a sphere of radius r2. Find the volume of the ring-shaped solid that remains. (b) Express the volume in terms of height (h).

### Metric geometry

In the taxi-cab plane show that ifA=(-5/2,2),B=(1/2,2), C=(2,2), P=(0,0), Q=(2,1) and R=(3,3/2)then A-B-C and P-Q-R. Show that line segment AB~to line segment PQ,line segment BC~line segmentQR, line segment AC~line segment PR. Sketch an appropriate picture.

### Equation of the tangent plane to the central conicoid

Problem 1 Find the equation of the tangent plane to the central conicoid x2 - 4y2 + 3z2 + 2 = 0 at the point (1,2,0). Problem 2 Find whether the plane 2x + 3y + 2z =3 touches the central conicoid 2x2 + 3y2 + z2 = 1 or not.

### Angles and their Link to the Number of Sides of a Polygon

Given the following measures of a vertex angle of a regular polygon, determine how many sides it has? a. 150 degrees b. 156 degrees c. 174 degrees d. 178 degrees

### Transformation geometry proof: reflections

Show \$_p*\$_l*\$_p*\$_l*\$_p*\$_l*\$_p is a reflection in a line parallel to line l note \$_p is a reflection about the line p