Difference between revisions of "10-327/Homework Assignment 1"

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'''Read''' sections 12 through 17 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 18 through 22, just to get a feel for the future.
 
'''Read''' sections 12 through 17 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, '''preread''' sections 18 through 22, just to get a feel for the future.
  
'''Solve and submit the following problems.''' In Munkres' book, problems 4, 8 on pages 83-84, problems 4, 8 on page 92, and problems 6, 7, 13 on page 101.
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'''Solve and submit the following problems.''' In Munkres' book, problems 4, 8 on pages 83-84, problems 4, 8 on page 92, problems 6, 7, 13 on page 101, and, for extra credit, the following problem:
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''Problem.'' Let <math>X</math> and <math>Y</math> be topological spaces and let <math>A\subset X</math> and <math>B\subset Y</math> be subsets thereof. Using only the definitions in terms of continuity of certain functions, show that the topology induced on <math>A\times B</math> as a subset of the product <math>X\times Y</math> is equal to the topology induced on it as a product of subsets of <math>X</math> and of <math>Y</math>. You are allowed to use the fact that two topologies <math>{\mathcal T}_1</math> and <math>{\mathcal T}_2</math> on some set <math>W</math> are equal if and only if the identity map regarded as a map from <math>(W, {\mathcal T}_1)</math> to <math>(W, {\mathcal T}_2)</math> is a homeomorphism. Words like "open sets" and "basis for a topology" are not allowed in your proof.
  
 
'''Due date.''' This assignment is due at the end of class on Thursday, September 30, 2010.
 
'''Due date.''' This assignment is due at the end of class on Thursday, September 30, 2010.
  
 
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Revision as of 13:16, 23 September 2010

In Preparation

The information below is preliminary and cannot be trusted! (v)

Read sections 12 through 17 in Munkres' textbook (Topology, 2nd edition). Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections 18 through 22, just to get a feel for the future.

Solve and submit the following problems. In Munkres' book, problems 4, 8 on pages 83-84, problems 4, 8 on page 92, problems 6, 7, 13 on page 101, and, for extra credit, the following problem:

Problem. Let X and Y be topological spaces and let A\subset X and B\subset Y be subsets thereof. Using only the definitions in terms of continuity of certain functions, show that the topology induced on A\times B as a subset of the product X\times Y is equal to the topology induced on it as a product of subsets of X and of Y. You are allowed to use the fact that two topologies {\mathcal T}_1 and {\mathcal T}_2 on some set W are equal if and only if the identity map regarded as a map from (W, {\mathcal T}_1) to (W, {\mathcal T}_2) is a homeomorphism. Words like "open sets" and "basis for a topology" are not allowed in your proof.

Due date. This assignment is due at the end of class on Thursday, September 30, 2010.

Dror's notes above / Student's notes below