Math 54 will be an introductory course in topology with an emphasis on developing geometric insight and in obtaining an overview of the various techniques used in topology (these include ideas from geometry, algebra and analysis). The topics covered in this course (and its sequel math 74) will include the fundamental concepts (continuity, compactness, connectedness, and product spaces), identification spaces, the fundamental group, topological groups, covering space theory, and the geometry and topology of compact surfaces.
More explicitly, in math 54 we will cover chapters 1 through 5 and 10.4 of Armstrong's Basic Topology as well as a proof of the classification of compact surfaces and some additional applications of the fundamental group. We will follow the following steps.
Step 1: Intuitive topology. We will learn to deal with "topological objects" by assuming their "obvious" properties as axioms. In the process we shall develop an understanding of the Euler Characteristic and classify compact connected surfaces (via Conway's ZIP proof).
Step2: Next we will formalize of intuitive topology via the language of set theory and explore the fundamental concepts (chapters 1 - 3) in terms of this language. This will include understanding carefully the concepts of continuity, basis, compactness, connectedness, and product spaces.
Step 3: Constructions. Now we will learn how to construct interesting topological spaces (in particular our initial surfaces). This will include identification spaces and a brief look into topological groups (chapter 4) and deck-like group actions.
At this point we will have our first exam .
Step 4: The fundamental group (chapter 5). We explore the fundamental group and learn to compute it via the deck theorem (using the language of covering space theory (10.4)).
Step 5: Applications. We will apply the fundamental group to understand a variety of basic topological questions. In particular we will prove the "meteorological" theorem, the "there's your scalp!" theorem, the Brower fixed point theorem, and several other results.
We will finish with our final exam .
In math 74 we will follow this syllabus.