10-327/Homework Assignment 1: Difference between revisions
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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. |
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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, |
'''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. |
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'''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 12:16, 23 September 2010
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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 [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be topological spaces and let [math]\displaystyle{ A\subset X }[/math] and [math]\displaystyle{ 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]\displaystyle{ A\times B }[/math] as a subset of the product [math]\displaystyle{ X\times Y }[/math] is equal to the topology induced on it as a product of subsets of [math]\displaystyle{ X }[/math] and of [math]\displaystyle{ Y }[/math]. You are allowed to use the fact that two topologies [math]\displaystyle{ {\mathcal T}_1 }[/math] and [math]\displaystyle{ {\mathcal T}_2 }[/math] on some set [math]\displaystyle{ W }[/math] are equal if and only if the identity map regarded as a map from [math]\displaystyle{ (W, {\mathcal T}_1) }[/math] to [math]\displaystyle{ (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.
| Dror's notes above / Student's notes below |
