Some multiple choice questions to play around with

I’ve been messing around with AI a bit and tried giving all of the lecture notes so far as well as the two written assignments to GPT o1 and asking it to produce some practice test questions. Overall, the results have been mixed, but when I asked it to produce simple multiple choice questions it seemed to do a good job. I just needed to throw away a few bad questions and fix a few others. Feel free to use these questions to test your understanding of the definitions but keep in mind they’re rather shallow, and the questions on the test will require deeper thought (and there will not be any multiple choice questions on the test).

I tried asking it to produce more complicated short and long answer questions but the results were significantly lower in quality and the solutions it produced were wrong more often than not. If you want some practice questions, the textbook has a lot of good exercises, many of which are worked out in the back of the book.

I read through all the questions and checked its answer key, but my experience with AI is that it is very good at making wrong things look convincing. So if you spot anything that looks off that I may have missed, please let me know!

One note: I am hesitant to involve AI in this course since I don’t want students to start to have doubts that maybe the assignments, lecture notes, or grading is done by AI. They are not. Even if I wanted to, we’re not quite there yet. If I ever distribute any material produced by AI in this course I will clearly say so.

I have no current plan to produce practice questions like this as a regular thing. I just wanted to get a sense of the current capabilities of AI.

Anyway, have fun! I hope you get some use from these!

1. Which of the following is not one of the axioms that a collection \(\tau\) of subsets of a set \(X\) must satisfy to be called a topology on \(X\)?

(a) \(X\) and \(\varnothing\) are in \(\tau\).
(b) Arbitrary unions of sets in \(\tau\) are in \(\tau\).
(c) Arbitrary intersections of sets in \(\tau\) are in \(\tau\).
(d) Finite intersections of sets in \(\tau\) are in \(\tau\).

2. Which of the following correctly describes the discrete topology on a set \(X\)?

(a) Every singleton set is closed but not open.
(b) Every subset of \(X\) is open.
(c) Only the empty set and \(X\) itself are open.
(d) A set is open if and only if its complement is finite.

3. In the cofinite topology on \(\mathbb{R}\), which of the following statements is true?

(a) Every set is open.
(b) A set is open if and only if it is a countable union of intervals.
(c) A set is open if and only if its complement is finite (or the set is empty).
(d) Only the empty set and \(\mathbb{R}\) are open.

4. Which of the following sets is a basis for the standard topology on \(\mathbb{R}\)?

(a) All finite subsets of \(\mathbb{R}\).
(b) All intervals of the form \([a,b)\).
(c) All intervals of the form \((a,b)\).
(d) All intervals of the form \((a,b]\).

5. Which of the following collections forms a basis for the subspace topology on a subset \(Y \subseteq X\)?

(a) All sets of the form \(V \cap Y\) where \(V\) is open in \(X\).
(b) All sets of the form \(Y \setminus F\) where \(F\) is finite.
(c) All sets of the form \(\emptyset\) and \(Y\).
(d) Only singleton sets \(\{y\}\) for \(y \in Y\).

6. Consider \(\mathbb{R}\) with the lower limit topology, generated by basis elements \([a,b)\). Which of the following statements is true?

(a) It is coarser than the standard topology on \(\mathbb{R}\).
(b) It is finer than the standard topology on \(\mathbb{R}\).
(c) It is the same as the standard topology on \(\mathbb{R}\).
(d) It is incomparable with the standard topology on \(\mathbb{R}\).

7. Let \(\tau\) and \(\tau'\) be topologies on the same set \(X\). We say \(\tau\) is finer than \(\tau'\) if:

(a) \(\tau\) has fewer open sets than \(\tau'\).
(b) Every \(\tau\)-open set is also \(\tau'\)-open.
(c) Every \(\tau'\)-open set is also \(\tau\)-open.
(d) \(\tau\) is incomparable with \(\tau'\).

8. Which of the following topologies on \(\mathbb{R}\) is Hausdorff?

(a) The cofinite topology.
(b) The trivial (indiscrete) topology \(\{\varnothing, \mathbb{R}\}\).
(c) The standard (Euclidean) topology.
(d) The upper topology.

9. In a Hausdorff space, which of the following is guaranteed?

(a) Any two distinct points have disjoint open neighborhoods.
(b) Any two distinct points have the same set of neighborhoods.
(c) A sequence has at most one limit point in the entire space.
(d) Both (a) and (c).

10. Suppose \(X\) has the discrete topology. Which of the following is not true?

(a) All subsets of \(X\) are open.
(b) All subsets of \(X\) are closed.
(c) The only dense subset is \(X\) itself.
(d) No sequence has a limit in \(X\).

11. The product topology on \(X\times Y\) (where \(X\) and \(Y\) are topological spaces) is generated by:

(a) All sets of the form \(U \cup V\) where \(U\subseteq X\) and \(V\subseteq Y\) are open.
(b) All sets of the form \((U \times Y) \cap (X \times V)\) where \(U,V\) are open in \(X\) and \(Y\).
(c) All sets of the form \(\{(x,y)\mid x\in U, y\in V\}\) where \(U\) is open in \(X\) and \(V\) is open in \(Y\).
(d) All sets of the form \(\{(x,x)\mid x\in X\}\).

12. Which of the following collections does not form a topological basis on \(\mathbb{R}\)?

(a) All open intervals \((a,b)\) with \(a<b\).
(b) All half-open intervals \([a,b)\) with \(a<b\).
(c) All open intervals \((q,r)\) with \(q,r\in \mathbb{Q}\).
(d) All intervals \((n,m)\) with \(n,m\in \mathbb{Z}\).
(e) They all do

(Recall the distinction between a basis for the standard topology vs. a basis for some topology.)

13. The subspace topology on \(Y\subseteq X\) is defined by:

(a) All sets \(V\subseteq Y\) such that \(V\) itself is open in \(X\).
(b) All sets of the form \(\varnothing\) and \(Y\).
(c) All sets of the form \(U\cap Y\) where \(U\) is open in \(X\).
(d) All subsets of \(Y\) whose complements in \(Y\) are finite.

14. Consider \(\mathbb{R}\) with the standard topology. The set \(\{(n,m)\mid n<m, n,m\in \mathbb{Z}\}\) (intervals with integer endpoints) does not form a basis for the standard topology. However, it does form a basis for:

(a) The discrete topology on \(\mathbb{R}\).
(b) Some coarser topology than the standard topology.
(c) Some finer topology than the standard topology.
(d) None of the above; it does not form a topological basis at all.

15. A continuous function \(f: X \to Y\) between topological spaces is characterized by:

(a) \(f\) is one-to-one.
(b) For every open set \(U\subseteq Y\), the preimage \(f^{-1}(U)\) is open in \(X\).
(c) For every closed set \(C\subseteq X\), the image \(f(C)\) is closed in \(Y\).
(d) For every open set \(U\subseteq X\), the image \(f(U)\) is open in \(Y\).

16. Let \(X\) and \(Y\) be topological spaces, and \(f: X\to Y\). The statement “\(f\) is continuous at \(x_0\)” means:

(a) \(f\) is continuous everywhere if it is continuous at \(x_0\).
(b) For every open neighborhood \(V\ni f(x_0)\), there is an open neighborhood \(U \ni x_0\) such that \(f(U)\subseteq V\).
(c) \(f\) is constant in some open neighborhood of \(x_0\).
(d) \(f\) is differentiable at \(x_0\).

17. If \(f:X\to Y\) and \(g:Y\to Z\) are continuous functions, then the composition \(g\circ f: X\to Z\) is:

(a) Not necessarily continuous.
(b) Always continuous.
(c) Continuous if \(g\) is a homeomorphism.
(d) Continuous if \(f\) is a bijection.

18. A homeomorphism between two topological spaces \(X\) and \(Y\) is a bijection \(f: X\to Y\) such that:

(a) \(f\) is continuous and its inverse \(f^{-1}\) is also continuous.
(b) \(f\) is continuous only.
(c) \(f\) is an isometry (preserves distance).
(d) \(f\) is monotonic.

19. Suppose \((x_n)\) is a sequence in a topological space \(X\). We say \((x_n)\) converges to \(x_\infty\) if:

(a) \(x_n = x_\infty\) eventually (i.e., for all large \(n\)).
(b) For every open set \(U\ni x_\infty\), there exists \(N\) such that \(x_n \in U\) for all \(n\ge N\).
(c) \(\|x_n - x_\infty\|\to 0\) as \(n\to\infty\).
(d) \(x_\infty\) is contained in infinitely many terms of the sequence.

20. Which of the following is a limit point of a set \(A\subseteq X\)?

(a) A point \(x\) such that some open neighborhood of \(x\) contains no points of \(A\).
(b) A point \(x\) such that \(x\not\in A\).
(c) A point \(x\) such that every open neighborhood of \(x\) meets \(A\) in a point other than \(x\) itself.
(d) A point \(x\) such that \(x\in A\).

21. The closure \(\overline{A}\) of a set \(A\subseteq X\) is defined as:

(a) \(\overline{A} = \bigcup\{U : U \text{ is open and } U\subseteq A\}\).
(b) \(\overline{A} = A \cup A'\), where \(A'\) is the set of all limit points of \(A\).
(c) \(\overline{A} = \varnothing\) if \(A\) is not dense.
(d) \(\overline{A} = \{x \mid x \in X\setminus A\}\).

22. A set \(A \subseteq X\) is dense in \(X\) if:

(a) \(A = X\).
(b) \(X\setminus A\) is infinite.
(c) \(\overline{A} = X\).
(d) \(A\) is open and closed in \(X\).

23. Let \(X=\mathbb{R}\) with the cofinite topology. A set \(A\subseteq X\) is dense in \(X\) if and only if:

(a) \(A\) is finite.
(b) \(A\) is infinite.
(c) \(\overline{A} = \varnothing\).
(d) \(\mathbb{R}\setminus A\) is infinite.

24. Let \(X\) be a topological space. A subset \(A\subseteq X\) is closed if and only if:

(a) \(X\setminus A\) is open.
(b) \(A=\varnothing\) or \(A=X\).
(c) \(X\setminus A\) is finite.
(d) \(A\) is an arbitrary intersection of open sets.

25. Which of the following is always true of closed sets in any topological space \(X\)?

(a) The intersection of any family (including infinite) of closed sets is closed.
(b) The union of any family of closed sets is closed.
(c) The union of finitely many closed sets is closed.
(d) Both (a) and (c).

26. A metric space \((X,d)\) is always:

(a) T1 (Fréchet).
(b) T2 (Hausdorff).
(c) Discrete.
(d) (a) and (b).

27. Which of the following metrics on \(\mathbb{R}^2\) does not generate the same topology as the standard Euclidean topology?

(a) \(d_1((x_1,y_1),(x_2,y_2)) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\).
(b) \(d_2((x_1,y_1),(x_2,y_2)) = |x_1-x_2| + |y_1-y_2|\).
(c) \(d_3((x_1,y_1),(x_2,y_2)) = \max\bigl(|x_1-x_2|, |y_1-y_2|\bigr)\).
(d) \(d_4((x_1,y_1),(x_2,y_2)) = \begin{cases}1 & \text{if }(x_1,y_1)\neq (x_2,y_2)\\ 0& \text{if }(x_1,y_1)=(x_2,y_2)\end{cases}\) (the discrete metric).