Topology Project - (Counter)examples in Topology
The field of topology aims to identify and classify topological spaces. In this project we will explore this question. We have already learned about some basic examples of topological spaces and some properties of these spaces. In the next few weeks we will be learning more about the properties of these spaces. For this project, you will chose a topological space from the book Counterexamples in Topology (available for you to borrow in the math office). You should analyze your space and determine its behavior with respect to each of the properties we discuss in class. This includes the following:

• description of the space and how it is constructed
• a picture or diagram of the space
• T_1 property
• Hausdorff property
• Connectedness
• Path connectedness
• Components
• Path components
• Locally connected
• Locally path connected
• Compactness
•
•
• ...more to come.

Suggested spaces include:
24 Uncountable Fort Space
29 Cantor Set - Ian
35 One Point Compactification of the Rationals - Hannah
48 Lexicographic Ordering on the Unit Square - Emily
51 Right Half-open Interval Topology - Lydia
52 Nested Interval Topology - Quinn
53 Overlapping Interval Topology - Ernie
57 Divisor Topology - Meghan
62 Double Pointed Reals - Katie
64 Smirnov's Deleted Sequence Topology - Ryan
73 Telophase Topology - Trent
74 Double Origin Topology - Beth
99 Maximal Compact Topology
116/117/118 Topologist's Sine Curve (pick one version) - Bert
119/120 Infinite Broom (pick one version) - Hayley

The final product of your project should be a research-style poster including as much as information as possible about your space and its properties. Where appropriate, proofs should be included. Your poster should be well-made, although it is not necessary to print it as a full formal poster. There are several examples of mathematical research posters in the hallways of Herak near the math office. You can look at these for reference on appropriate style considerations. Posters will be presented to the class during our scheduled final. We will break the class time into two sessions, and you will be asked to stay near your poster and answer questions for one of these two sessions.

Your grade on the project will be based on:

• The correctness of the mathematical content
• The completeness of your work with respect to the project description
• The organization of your poster and presentation of your results
• Your ability to communicate your results and answer questions during the poster session