“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:

- a short (2-3 sentence) summary of what you read, and
- 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:

- 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.
- 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!*)

For those of you who want to more motivation for the quotient topology, here’s a page including several nice examples (and pictures!).