Reading 2 – Topology

TopologyExamples

“Point set topology is a disease from which the human race will soon recover.”

—Henri Poincaré, father of modern topology

The goal of your second reading assignment is to make sure you understand at least the basic definition of each object we discussed in the last lecture, namely:

Topological Space
Metric / Metric Topology
Equivalence Relation / Quotient Topology

You may of course use whatever literature you like, since these definitions are all fairly standard. You should also read the Wikipedia page on topology, which gives a nice overview of its history and applications.

Submission: As usual, please send an email to kmcrane@cs.cmu.edu and nsharp@cs.cmu.edu no later than 10:00 AM on Thursday, January 21st including the string DDGSpring2016 in your subject line.  Your email for readings should always include:

  1. a short (2-3 sentence) summary of what you read, and
  2. at least one question about something you found confusing / incomplete / not addressed.

To test your own understanding, your email summary should answer the following two questions:

  1. Treating the black dots as points, and the blue blobs as subsets, which of the collections A–F is a topological space? Note that a blob with no points in it is the empty set.
  2. Two points \(x\) and \(y\) are “separated by neighborhoods” if there is a set \(U\) containing \(x\) and a set \(V\) containing \(y\) such that \(U \cap V = \emptyset\). Suppose we define an equivalence relation \(x \sim y\) where \(x\) is equivalent to \(y\) if and only if \(x\) and \(y\) cannot be separated. Again referencing the picture above, what then is quotient topology \(E/\!\sim\)? (Hint: this topology has a special name!)

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