Point-set topology: exercises

These are exercises for Point-set topology: course.

1. Prove that the space of continuous functions $f:[0,1]\rightarrow {\bf R}$ is a metric space with the metric $d(f,g)=\max |f(x)-g(x)|$.
2. Is the boundary of a set always closed?
3. Show that the addition, subtraction, and multiplication operations are continuous functions from ${\bf R} \times {\bf R}$ into ${\bf R}$; and the quotient operation is a continuous function from ${\bf R} \times ({\bf R} \backslash \{0\})$ to ${\bf R}$. Show that the sum and the product of continuous functions (into ${\bf R}$) is continuous.
4. Is $D(E,E')=\max \{\min \{d(x,y),1\}:x \in E, y \in E'\}$ a metric on the quotient space?
5. Let $f_n:X\rightarrow Y$ be a sequence of continuous functions uniformly convergent to $f$. Let $x_n$ be a sequence of points of $X$ converging to $x$. Show that $f_n(x_n)$ converges to $f(x)$. What happens when the convergence is not uniform?
6. Show that $(X,d)$ is totally bounded iff $(X,d')$ is totally bounded, where $d'(x,y)=\min \{d(x,y),1\}$.
7. Find all possible topologies on the set of $3$ elements.
8. Suppose $F$ is closed and $G$ is open. Show that $F \backslash G$ is closed and $G \backslash F$ is open.
9. Show that any closed convex polyhedron is homeomorphic to a closed ball in ${\bf R}^3$.
10. Show that if $X$ is regular, every pair of points of $X$ have neighborhoods whose closures are disjoint.
11. Show that the Tietze extension theorem implies the Urysohn lemma.
12. Show that if $X\times Y$ is connected and nonempty then both $X$ and $Y$ are connected.
13. Show that $X=\text{Graph}\{\sin (1/x)\}\cup (\{0\}\times [-1,1])$ is connected but not path-connected.
14. Show that if $Y$ is compact then the projection $p:X\times Y\rightarrow X$ is a closed map
15. Let $f:X\rightarrow Y$ be a function, $Y$ compact Hausdorff. Prove that $f$ is continuous iff the graph of $f$ is closed in $X\times Y$.
16. Prove that every regular space with a countable base is normal.
17. Prove that for any set $A\subset\mathbf{R}^{n},$ $Fr(A)$ is closed, i.e., its complement is open.
18. Show that any set is both open and closed relative to itself.
19. Show that if $X$ is a non-empty topological space with the discrete topology, then the only connected sets are the sets of one element.
20. Prove that $[0,1]$ is connected. (You can use the fact that it is compact.)
21. Prove that a closed subset of a compact topological space is compact.
22. Show that the standard Euclidean (disk) basis of $\mathbf{R}^{2}$ is equivalent to the basis of the product topology of $\mathbf{R}^{2}=\mathbf{R}\times\mathbf{R}.$
23. Let $X=\mathbf{S}^{1}=\{(x,y):x^{2}+y^{2}=1\}$ (the unit circle) and $(x,y)\sim(-x,-y)$ for all $(x,y)\in X.$ Describe $X/_{\sim}.$
24. Prove that neighborhoods are open.
25. Is the union of a collection of closed sets always closed?
26. Prove that the frontier is closed.
27. Suppose $A$ is a subset of a topological space $X$ and $τ$ is the topology of $X$. Define a collection of subsets of $A$ as $τ_A = \{W∩A: W∈τ\}$. Prove that the union of any subcollection of $τ_{A}$ belongs to $τ_{A}$.
28. Prove that $f(x) = x²$ is continuous at $x=0$.
29. (a) Prove that the projection $p: {\bf R}^2 → {\bf R}$ is continuous. (b) Prove that the projection of a (filled) square on one of its sides is continuous.
30. Prove that a function is continuous if and only if the preimage of any closed set is closed.
31. (a) Prove that all open intervals of finite length are homeomorphic. (b) Are $(0,1)$ and $(0,∞)$ homeomorphic?
32. True or false?
    1. If $X×Y$ is path-connected then so is $X$ and $Y$.
    2. In ${\bf R}^n$, an unbounded set is not compact.
    3. The empty set is compact.
33. Give an example of:
    1. a continuous function $f:X→X\times Y$ which isn't constant,
    2. non-compact, bounded subset of ${\bf R}^2$ which isn't open,
    3. a projection $p:X\times Y \to X$ which isn't continuous,
    4. a non-Hausdorff topology on ${\bf R}^2$ which isn't anti-discrete.
34. (a) Give the definition of a Hausdorff space. (b) Prove that a subspace of a Hausdorff space is Hausdorff. (c) State the theorem about homeomorphisms of Hausdorff spaces.
35. (a) Define the product of topological spaces. (b) Prove that the product of two path-connected spaces is path-connected. (c) State the theorem about products of compact spaces.
36. Let $f:X\rightarrow X$ be continuous. If $f(x)=x$ then $x$ is called a fixed point of $x.$ Show that $f$ always has a fixed point when $X=[0,1].$ What happens when $X=[0,1)?$
37. Show that every compact subset of a metric space is bounded and closed.
38. Let $f:{\bf R}^n \rightarrow {\bf R}^m$ be continuous and $M\subset R^{n}$ be path-connected. Show that $f(M)$ is path-connected.