Math · 09
Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
In \(\mathbb{R}\) with the standard topology, which of \((0,1)\) and \([0,1]\) is compact? Give a cover that kills the one that fails.
Timer not started. Twenty minutes of real work before a solution.
\([0,1]\) is compact by Heine–Borel. \((0,1)\) is not: the cover \(\{(1/n, 1): n\ge 2\}\) has no finite subcover, because any finite subcollection has a largest \(n\) and then misses points near 0.
Attack 2
Is \(\mathbb{Q}\), with the subspace topology from \(\mathbb{R}\), connected?
Timer not started. Twenty minutes of real work before a solution.
No. \(\mathbb{Q}\cap(-\infty,\sqrt{2})\) and \(\mathbb{Q}\cap(\sqrt{2},\infty)\) are nonempty, disjoint, open in \(\mathbb{Q}\), and cover \(\mathbb{Q}\).
Attack 3
Prove that the continuous image of a connected space is connected.
Timer not started. Twenty minutes of real work before a solution.
If the image split into two nonempty sets open in the image, their preimages would split the domain into two nonempty sets open in the domain. So a connected domain cannot map onto a disconnected space.
Point-set topology is what remains of analysis when you delete the metric and keep the language of nearness. An open set is no longer a union of balls; it is a member of a declared collection closed under arbitrary unions and finite intersections. Continuity becomes “the preimage of an open set is open.” Compactness becomes “every open cover has a finite subcover,” which on \(\mathbb{R}^n\) with the standard topology is the Heine–Borel theorem you already used without naming. Connectedness, separation axioms, and countability axioms are the other switches. The fundamental group is the first algebraic invariant: loops up to continuous deformation.
Munkres, Topology, 2nd edition, ISBN 978-0-13-181629-9, is the main text and it is paid. The supplementary exercises are the ceiling. MIT 18.901 from fall 2004 posts problem sets keyed to those sections; that page is your plan. Topology Without Tears is the free book if you need a gentler parallel text. π-Base is a database of spaces and counterexamples, useful after you have a definition, useless as a first reading. Learn Anything’s topology page is mostly journalism and applied-topology ads. The Quanta piece below is the one preview worth the slot. The Munkres solution repository is a check you run after your writeup, not a teacher.
A space is a list of open sets plus the theorems you can prove from that list. If you cannot produce a counterexample when a hypothesis is dropped, you do not know the theorem.
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
Spaces and bases. A topology on \(X\) is a collection of subsets containing \(\emptyset\) and \(X\), closed under arbitrary unions and finite intersections. A basis generates a topology by taking arbitrary unions. The subspace, product, and quotient topologies are defined by the coarsest or finest topology making the relevant maps continuous, and you should be able to write each one with open sets.
These are definitions. The lemma you must prove: a basis satisfies the intersection condition, and the unions form a topology. Product topology: basic open sets are products of opens with all but finitely many factors equal to the whole space, in the case of arbitrary products; for two spaces it is products of opens.
Continuity and separation. \(f\) is continuous if and only if preimages of opens are open. Hausdorff: distinct points have disjoint neighborhoods. Metric spaces are Hausdorff. Compact subsets of Hausdorff spaces are closed.
The preimage criterion matches the metric ε–δ definition when the topology comes from a metric, because balls form a basis. Hausdorff plus compact implies closed: separate a point outside from every point inside, take a finite subcover of the inside, and intersect the corresponding outside neighborhoods.
Connectedness and compactness. Connected means the space is not the union of two nonempty disjoint opens. Path-connected implies connected. Compact means every open cover has a finite subcover. Continuous images preserve both. A closed subset of a compact space is compact. In \(\mathbb{R}^n\), compact iff closed and bounded.
Preservation under continuous maps is the preimage argument from the attack. Closed subsets of compact sets: add the complement to a cover and throw it away after taking a finite subcover. Heine–Borel is the one hard theorem; prove it for \([a,b]\) by the least-upper-bound argument, then for boxes, then for closed bounded sets by sitting inside a box.
Countability and metric spaces. First countable: countable local bases. Second countable: countable basis for the whole topology. Second countable implies separable and Lindelöf. A metric space is first countable. Urysohn’s metrization theorem: a second-countable regular Hausdorff space is metrizable. You must be able to state it and know why each hypothesis is there, even if the proof is a project.
Metric first countability: balls of radius \(1/n\). Second countability implies separable by picking a point from each basis element. Counterexamples (an uncountable discrete space, the Sorgenfrey line’s square) are the content of the exercises in Munkres. Build two of them.
Fundamental group. Loops based at \(x_0\), modulo path homotopy, form a group \(\pi_1(X,x_0)\). A continuous map induces a homomorphism. \(\pi_1(S^1)\cong\mathbb{Z}\). A covering map induces an injection on fundamental groups. Simply connected means path-connected and trivial fundamental group.
The group operation is concatenation of paths; homotopy of paths supplies the inverse and associativity up to homotopy. The isomorphism \(\pi_1(S^1)\cong\mathbb{Z}\) is the winding-number argument via the covering \(\mathbb{R}\to S^1\). Write that lifting argument once, slowly.
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Watch or read once. Do not study from this.
Quanta, Topology 101
Read once, do not study from this. It is journalism about holes. It is also the only Learn Anything topology link that is a genuine preview rather than a paper or a tool. Then leave it.
The thing you work through.
Munkres, Topology, 2nd edition, ISBN 978-0-13-181629-9, read beside Topology Without Tears
Munkres is paid and is the main text; the ISBN is the catalogued 2nd edition. Topology Without Tears is the free parallel book when a Munkres paragraph is too compressed. Use both. Do not replace Munkres’s exercises with the free book’s.
The problem book.
MIT 18.901, Fall 2004, assignments
The problem plan, with section and exercise numbers aimed at Munkres. Do these in order. The course page is https://ocw.mit.edu/courses/18-901-introduction-to-topology-fall-2004/ .
The hardest questions that exist at this level.
Munkres supplementary exercises, checked only after your writeup
The repository is a solution check, not a text. Open it after the 20-minute timer and a complete writeup, one exercise at a time. π-Base is the counterexample catalog when you need a space: https://topology.pi-base.org/ .
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: the 18.901 assignment list, completed in order through compactness and the fundamental group, with written solutions. You must also prove, from memory, that a continuous image of a compact space is compact, and sketch why π₁(S¹) is ℤ. Fail if “open set” still means “blob” rather than “member of the topology.”
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
Define a topology, define compactness, prove that continuous images of compact spaces are compact, and say what the fundamental group measures. Blank page.