For someone who is self-studying topology: what are the main topics to focus on?

Question

I will have to teach myself topology for the Math GRE Subject Test because, although I graduated with a math major, I never took topology.

I have Munkres and Kelley, along with the Schaum's Outlines and am intimidated by the amount of material there is, and given my limited amount of time, I need to know where to focus. For example, I imagine that understanding what a topological space is more important to understanding Topology than understanding partial orderings, but I have no idea if I need to know partial orderings to understand something later.

What topics are the most important in an undergraduate topology course? Chapter mappings with respect to Munkres textbook would be especially appreciated.

If this question is too broad or inappropriate for this site, I can remove this question.

Answer 1

I think this is a reasonable question.

If your purpose is to study for the math GRE, you need to know very little. The first 3 chapters of Munkres contains far more than you need for the test, for instance. The classic responses that the math GRE is a mile wide, but an inch deep and that a good way to understand what topics are included is to take several practice tests. Particularly important concepts for the test include

1. Knowing what a topology is
2. Knowing the basic topologies, like the discrete, quotient, product, and (most importantly) the metric topology
3. Knowing some fundamental topological properties, like connectedness and compactness

These are often included in real analysis texts, and having that level of understanding is about right for the test.

If your purpose is to go on to learn higher mathematics, you will come across topology all the time. So often that it will be assumed and often unmentioned, much like how calculus is used in higher math or physics. It's used implicitly all the time, and assumed so well known that explanation would be a waste of time. It would be to your benefit to have a good understanding of topology before entering grad school.

Answer 2

You should learn a healthy amount of set theory and an intermediate amount of analysis. Knowing set algebra (union, intersection, difference, showing two sets are equal, etc.) will be absolutely necessary for your success in Topology. Once you get through the basics of Topology, you'll find that functions (usually continuous or uniformly continuous) between spaces will become a very important concept. Continuity in Topology is cast in terms of open sets. This is where you'll want to have some real analysis to fall back on, as it will be very useful if you already have a rigorous understanding of $\varepsilon$-$\delta$ continuous functions. It would also be very useful to know about sequences and convergence of sequences

In short, you can probably get through some basic Topology with an intermediate amount of set theory under your belt (you should still know sequences). To go further, I'd recommend knowing some analysis to help you with functions in Topological spaces. In my own experience, I could've learned all the Topology I know without once discussing partial orderings.