Topology
2nd Edition
ISBN: 9780134689517
Author: Munkres, James R.
Publisher: Pearson,
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Chapter 4.30, Problem 14E
Show that if
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2) Suppose you start with $60 and increase this amount by 15%. Since 15% of $60 is $9,
that means you increase your $60 by $9, so you now have $69. Notice that we did this
calculation in two steps: first we multiplied $60 by 0.15 to find 15% of $60, then we
added this amount to our original $60. Explain why it makes sense that increasing $60 by
15% can also be accomplished in one step by multiplying $60 times 1.15.
3) Suppose you have $60 and want to decrease this amount by 15%. Since 15% of $60 is $9,
that means you will decrease your $60 by $9, so you now have $51. Notice that we did this
calculation in two steps: first we multiplied $60 by 0.15 to find 15% of $60, then we
subtracted this amount from our original $60. Explain why it makes sense that decreasing
$60 by 15% can also be accomplished in one step by multiplying $60 times 0.85.
4) In the Read and Study section, we noted that the population in Colony B is increasing each
year by 25%. Which other colony in the Class Activity…
5) You are purchasing a game for $30. You have a 5% off coupon and sales tax is 5%. What
will your final price be? Does it matter if you take off the coupon first or add in the tax first?
6) You have ten coupons that allow you to take 10% off the sales price of a jacket, and for
some strange reason, the store is going to allow you to use all ten coupons! Does this mean
you get the jacket for free? Let's really think about what would happen at the checkout.
First, the teller would scan the price tag on the jacket, and the computer would show the
price is $100. After the teller scans the first coupon, the computer will take 10% off of
$100, and show the price is $90. (Right? Think about why this is.) Then after the teller scans
the second coupon, the computer will take 10% off of $90.
(a) Continue this reasoning to fill in the table below showing the price of the jacket (y) after
you apply x coupons.
(b) Make a graph showing the price of the jacket from x = 0 to x = 10 coupons applied.…
(a)
(b)
(c)
(d)
de
unique?
Answer the following questions related to the linear system
x + y + z = 2
x-y+z=0
2x + y 2 3
rewrite the linear system into the matrix-vector form A = 5
Fuse elementary row operation to solve this linear system. Is the solution
use elementary row operation to find the inverse of A and then solve
the linear system. Verify the solution is the same as (b).
give the null space of matrix A and find the dimension of null space.
give the column space of matrix A and find the dimension of the column
space of A (Hint: use Rank-Nullity Theorem).
Chapter 4 Solutions
Topology
Ch. 4.30 - Show that l and I02 are not metrizable.Ch. 4.30 - Which of our four countability axioms does S...Ch. 4.30 - Which of our four countability axioms does in the...Ch. 4.30 - Let A be a closed subspace of X. Show that if X is...Ch. 4.30 - Prob. 10ECh. 4.30 - Let f:XY be continuous. Show that if X is...Ch. 4.30 - Let f:XY be continuous open map. Show that if X...Ch. 4.30 - Show that if X has a countable dense subset, every...Ch. 4.30 - Show that if X is Lindelof and Y is compact, then...Ch. 4.30 - Give I the uniform metric, where I=[0,1]. Let...
Ch. 4.30 - Prob. 16ECh. 4.30 - Prob. 17ECh. 4.30 - Prob. 18ECh. 4.31 - Show that if X is regular, every pair of points of...Ch. 4.31 - Show that if X is normal, every pair of disjoint...Ch. 4.31 - Show that every order topology is regular.Ch. 4.31 - Prob. 4ECh. 4.31 - Prob. 5ECh. 4.32 - Which of the following spaces are completely...Ch. 4.32 - Prob. 8ECh. 4.32 - Prove the following: Theorem: If J is uncountable,...Ch. 4.32 - Prob. 10ECh. 4.33 - Examine the proof of the Urysohn lemma, and show...Ch. 4.33 - a Show that a connected normal space having more...Ch. 4.33 - Give a direct proof of the Urysohn lemma for a...Ch. 4.33 - Prob. 4ECh. 4.33 - Prob. 5ECh. 4.33 - Prob. 8ECh. 4.34 - Give an example showing that a Hausdorff space...Ch. 4.34 - Give an example showing that a space can be...Ch. 4.34 - Let X be a compact Hausdorff space. Show that X is...Ch. 4.34 - Let X be a locally compact Hausdorff space. Is it...Ch. 4.34 - Let X be a locally compact Hausdorff space. Let Y...Ch. 4.34 - Check the details of the proof of Theorem 34.2.Ch. 4.34 - A space X is locally metrizable if each point x of...Ch. 4.34 - Show that a regular Lindelof space is metrizable...Ch. 4.35 - Show that the Tietze extension theorem implies the...Ch. 4.35 - In the proof of the Tietze theorem, how essential...Ch. 4.35 - Let X be metrizable. Show that the following are...Ch. 4.35 - Let Z be a topological space. If Y is a subspace...Ch. 4.35 - Prob. 5ECh. 4.35 - Let Y be a normal space. The Y is said to be an...Ch. 4.35 - a Show the logarithmic spiral...Ch. 4.35 - Prove the following: Theorem. Let Y be a normal...Ch. 4.36 - Prove that every manifold is regular and hence...Ch. 4.36 - Let X be a compact Hausdorff space. Suppose that...Ch. 4.36 - Let X be a Hausdorff space such that each point of...Ch. 4.36 - Prob. 5ECh. 4.SE - Consider the following properties a space may...Ch. 4.SE - Consider the following properties a space may...Ch. 4.SE - Prob. 3SECh. 4.SE - Consider the following properties a space may...
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