Tag Archives: study problems

Some study questions (in no particular order)

Leave a comment Posted by on March 5, 2015

1. Let U \subset R^2 . Declare U to be open in topology \mathscr{T}_r if the following holds: given (x,y) \in U and radian angle \theta \in [0, 2 \pi) there exists \delta_{\theta} > 0 such that the open line segment r(t) = \langle x,y \rangle + t \langle cos(\theta), sin(\theta) \rangle \delta_{\theta}, t \in [0,1) is also in U . That is, for all (x,y) is U there is, for every direction, some open segment starting at (x,y) that stays in U . This topology is described in Willard in problem 3A, part 4 (“radially open”).

Is this topology finer, coarser or incomparable to the usual topology of the plane?

(hint: consider (0,0) and consider the set (described in polar coordinates): r < e^{-\frac{1}{2 \pi -\theta}}, \theta \in [0, 2 \pi) : is this set open in the radial topology? Is it open in the usual topology?

2. Let A \subset R^1 in the usual topology. Assume that A is non-empty. Let a = lub\{x \in A \} . If A is closed, is it always true that a \in A ? If A is open, is it always true (or EVER true) that a \in A ?

3. Same questions as 2, but now assume that R has the “lower limit” topology (open sets are generated by [c,b), b > c ).

4. Let f:(X, \mathscr{T}_1) \rightarrow (Y, \mathscr{T}_2) be a continuous, onto function between Hausdorff topological spaces. Endow X \times Y with the product topology. Prove (or provide a counter example): \Gamma = \{(x,y), | x \in X, y = f(x) \} is a closed set in X \times Y .

5. Given R^Z (countable product of the real line), describe a set that is open in the box topology that is not open in the product topology.

6. Given f: X \rightarrow Y is continuous and C \subset Y is closed, prove that f^{-1}(C) is closed in X .

If X is a topological space and a_n is a sequence of points in X , we say that l \in X is a limit point for the sequence if, for every open set U, l \in U , there exists at least one a_n \neq l where a_n \in U . We say that the sequence converges to a if, for every open set U, a \in U , there is some number M such that, for all n \geq M, a_n \in U .

7. Suppose that f: X \rightarrow Y is continuous. Then if x_n \rightarrow x, f(x_n) \rightarrow f(x) .

8-11. Let a_n = \frac{1}{n}, b_n = \frac{n}{n+1}, c_n = (-1)^n b_n

8. For each sequence a_n, b_n, c_n , find the limit points and if the sequence converges if R has the usual topology.

9. For each sequence a_n, b_n, c_n , find the limit points and if the sequence converges if R has the lower limit topology.

10. For each sequence a_n, b_n, c_n , find the limit points and if the sequence converges if R has the discrete topology.

11. For each sequence a_n, b_n, c_n , find the limit points and if the sequence converges if R has the finite complement topology.

12. Show that a sequence can have at most ONE limit point if X is Hausdorff (has the T_2 property). Show that this is false if the topology is not assumed to be Hausdorff.

13. Let X be a Hausdorff topological space. Show that a one point set is a closed set. Is this true of the space is merely T_1 ?

14. Let Y be a Hausdorff topological space. Let f:X \rightarrow Y and g:X \rightarrow Y be continuous functions. Show: if S = \{ x| f(x)=g(x) \} is a closed set.

15. Show that the comb spaces and the topologist’s sine curve are connected sets (in the usual subspace topology of R^2 ).

16. Show that R with the usual topology, the lower limit topology, the discrete topology and the finite complement topology are NOT topologically equivalent spaces.

17. Show that the unit circle (usual subspace topology) and the unit interval are not topologically equivalent spaces.

18. Show that the “closed unit interval” is not a connected set in the lower limit topology.

19. Show that the following set \{ (x,y), y = x \sin(\frac{1}{x}, x \in (0, \pi] \} \cup \{(0,0) \} is both a connected set and a path connected set (usual subspace topology).

20. Let X have topologies \mathscr{T}_a, \mathscr{T}_b . Suppose \mathscr{T}_a \subset \mathscr{T}_b (that is the topology \mathscr{T}_b is finer than the topology \mathscr{T}_a . Show that if (X, \mathscr{T}_a ) is Hausdorff, then so is (X, \mathscr{T}_b) . Does it work the other way? Why or why not?

21. Consider the following “sequence of sequences” in R^{Z} : x_1 = (1, 0, 0, ...), x_2 = (0, \frac{1}{2}, 0, 0, 0,..), x_3 = (0, 0, \frac{1}{3}, 0, 0,...) , x_n is the point that is \frac{1}{n} in the n’th coordinate, and 0 in the other coordinates. That is, \pi_k (x_n) = \frac{1}{k} if k=n and zero otherwise.

Does x_n converge to (0, 0, 0,.....0, ) in the product topology? What about in the box topology?

22. Suppose A \subset X is closed. Show that Fr(A) \subset A .

23. Suppose that for all i \in \{1, 2, 3, ...\} we have that X_i is a Hausdorff topological space (note: the X_i might not be topologically equivalent spaces).

Show: \Pi^{\infty}_{i}X_i is Hausdorff in the product topology. Is this true if we use the box topology?

24. Show that if A, B are connected subsets of X and A \cap B \neq \emptyset then A \cup B is connected.

25. Show that if A is a connected set, then so is \bar{A} . If \bar{A} is connected, does it follow that A is connected?

26. Show that if X, Y are connected topological spaces, then so is X \times Y .

27. Show that if there exists a non-empty subset A \subset X where A \neq X and A is both open and closed, then X is NOT a connected space.

28. Let A_n denote a set with precisely n elements. For n \in {2, 3, 4, 5, 6} , find all possible topologies for A_n .

29-30: A topological space X is called “topologically homogeneous” if, given any two points x, y \in X , there is a homeomorphism h: X \rightarrow X where h(x) = y .

29. Are the following topological spaces (usual topology) topologically homogeneous: [0,1] \subset R , (0,1) \subset R , S^1 \subset R^2 , “the wedge of 2 circles (last homework)” ?

30. Prove that R^n (usual topology, n finite) is topologically homogeneous.

connected sets, continuous functions, metric spaces, open set, path connected, Separation Axioms, sequence, subspace topology, topologies