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I know pretty much nothing about topology except that there are closed loops on a torus's surface that can't be continuously deformed into a point. What is the best possible accessible question for a beginner to try to solve to learn more about topology?

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"Where can I get a copy of Topology by Munkres?" – user31373 Jun 21 '12 at 2:27
This is highly subjective. Best possible for whom? (Do you mean point-set topology? Geometric topology? Algebraic topology?) – Qiaochu Yuan Jun 21 '12 at 2:44
@QiaochuYuan Best possible for me, i.e. approximately someone who knows how to determine whether a statement is true or false. As far as sub-branches of topology go, I know nothing about them. – idonutunderstand Jun 21 '12 at 4:09
What is the fundamental group of the torus? – i. m. soloveichik Aug 8 '12 at 3:34

For a very lively and readable introduction to the ideas of topology, take a look at The Shape of Space by Jeff Weeks. This book is at a level suitable for a good high-school student, and introduces many of the basic ideas of the subject in an entertaining way.

As far as problems go, here are a few basic ones:

  1. As you are probably aware, you can obtain a torus by gluing together the opposite sides of a square. What surface do you get if you glue together the opposite sides of a hexagon? What about an octagon?

  2. You are probably aware of the Möbius strip, a surface which has only one side. What surface do you get if you cut a Möbius strip along the center line? Does it disconnect the strip? What surface do you get if you glue together two Möbius strips along their respective "edges"?

  3. As you may have heard, the water, gas, electricity graph ($K_{3,3}$) cannot be drawn on the plane without crossings. Show that it can be drawn on the torus without crossings. Can every graph be drawn on the torus without crossings?

  4. Find a way to connect three rubber bands so that they cannot be pulled apart, but if you cut any one of the three, then all three can be pulled apart.

If you are serious about learning topology and have a background in proof-based mathematics, the best books to look at are probably Topology of Surfaces by Kinsey and Topology by Munkres. The first is more geometric, while the second is an introduction to the formalism of point-set topology.

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If you want to connect back to things you've learned in Calculus, I'd suggest proving the Intermediate Value Theorem. The proof of the Intermediate Value Theorem, which states

Given a continuous function $f:\mathbb R\to \mathbb R$ and points $a,b\in \mathbb R$, if $f(a)\geq x\geq f(b)$ or $f(b)\geq x\geq f(a)$ then there exists some $c\in [a,b]$ such that $f(c)=x$.

is quite topological in nature (hint: it uses connectedness). On the note of connectedness, I'd also show that these four definitions of connectedness are equivalent:

A space $X$ is connected if it is not the union of two disjoint open subsets.

A space $X$ is connected if it is not the union of two disjoint closed subsets.

A space $X$ is connected if the only subsets which are both open and closed are $\emptyset$ and $X$.

A space $X$ is connected if any continuous function $f:X\to \{0,1\}$ is constant.

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A couple of years ago I wrote up a very short introduction to the ideas of topology. The exercises are typical, but fairly easy. It is unfinished, and still contains a number of errors, but it might be worth a look anyway:

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Prove that a subset of $\mathbb R$ is closed if and only if it is the complement of an open set. The definition of a closed set is as follows:

A set $C$ is closed if every limit point of $C$ is contained in $C$. A point $p$ is a limit point of some set $S$ if every open set containing $p$ intersects $S\setminus\{p\}$.

Open sets are the fundamental concept of topology. They satisfy the following axioms: $\mathbb R$ is an open set. The intersection of any two open sets is open. The union of any collection of open sets is open.

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Here's a nice question that should test the very basics of topology.

  1. Given any topological space $X$, prove that starting with any subset $A \subseteq X$ at most fourteen distinct sets can be obtained by repeated applications of the operations "complement" and "closure". I.e., if $A^*$ denotes the complement $X \setminus A$ of $A$, and $A^-$ denotes the closure $\overline{A}$ of $A$, then among $A, A^*, A^-, A^{**}, A^{*-}, A^{--}, A^{-*}, \ldots$ there are at most fourteen different sets.
  2. Find a subset $A$ of the real line from which one obtains fourteen different sets in this manner.
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Another se.math user asked recently on chat for a not-too-hard topology exercise, and I leafed through J.L. Kelley's General Topology (1955) and found this one, exercise 1.D(a) on page 56:

A topological space is a $T_1$-space if each set which consists of a single point is closed. Show that for any set $X$ there is a unique smallest topology $\mathfrak I$ such that $(X, \mathfrak I)$ is a $T_1$-space.

It is not a difficult exercise, but I think it is a good one because you can't solve it unless you have a firm grasp on some of the essentials of topology.

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Here is quite a hard problem. There is a theorem that any embedding of $K_7$ in $3$-space contains a knot. So draw such an embedding by putting 7 points on a piece of paper and join each to every other one with overpasses and underpasses. Then find a knot in this, i.e. a loop through all 7 points on your graph which is knotted. (There may be more than one in some cases.)

Another result is that any embedding of $K_6$ in $3$-space contains a link, and this is usually not so hard to find.

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