Math 221A homework

Due 10/13

In this homework, $X$ and $Y$ are nonempty topological spaces.

  1. [pointwise open] Prove that $U$ is open if and only if for all $x \in U$ there exists an open set $U_x$ such that $x \in U_x \subseteq U$.
  2. [projections are open] A map $f$ is called an open map if $f(U)$ is open for every open subset $U$ of the domain. Prove that the projection $\pi_1 \colon X \times Y \to X$ is an open map.
  3. [closure $\times$ closure] Prove that if $A \subseteq X$ and $B \subseteq Y$ then $\overline{A \times B} = \overline{A} \times \overline{B}$.
  4. [closure $\cup$ closure] Prove that if $A$ and $B$ are subsets of $X$ then $\overline{A \cup B} = \overline{A} \cup \overline{B}$.
  5. [$\bigcup$ closure] Prove that if $A_\alpha$ is a collection of subsets of $X$ then $\bigcup \overline{A_\alpha} \subseteq \overline{\bigcup A_\alpha}$, but equality does not necessarily hold.
  6. [Hausdorff $\times$ Hausdorff] Prove that if $X$ and $Y$ are Hausdorff then so is $X \times Y$.
  7. Prove that $X$ is Hausdorff if and only if $\{(x,x) \mid x \in X\}$ is a closed subset of $X \times X$.
  8. The boundary of a set $A$ is defined to be $\text{Bd }A = \overline{A} \cap \overline{X \setminus A}$. Prove that the closure of $A$ is the disjoint union of $\text{Int }A$ and $\text{Bd }A$.

Due 10/20 — no, make that 10/23

Let $X$ and $Y$ be nonempty topological spaces. Let $X_\alpha$ be a family of nonempty topological spaces, and give $\Pi X_\alpha$ the product topology.

  1. Prove that there is an embedding of $X$ into $X \times Y$.
  2. Prove that every open interval in $\mathbb{R}$ is homeomorphic to $\mathbb{R}$.
  3. Give an example of a function $\mathbb{R} \to \mathbb{R}$ that is continuous at exactly one point.
  4. Suppose $Y$ is Hausdorff, and $f \colon X \to Y$ and $g \colon X \to Y$ are continuous functions. Prove that if $f|_A = g|_A$ for some dense set $A$ then $f = g$. (Here, dense means $\overline{A} = X$.)
  5. Prove that if $A_\alpha$ is a closed subset of $X_\alpha$ for all $\alpha$ then $\Pi A_\alpha$ is a closed subset of $\Pi X_\alpha$.
  6. Suppose $\vec{y}$ is a point in $\Pi X_\alpha$ and $(\vec{x_n})$ is a sequence of points in $\Pi X_\alpha$. Prove that the sequence $\vec{x_n}$ converges to $\vec{y}$ if and only if the sequence $\pi_\alpha(\vec{x_n})$ converges to $\pi_\alpha(\vec{y})$ for all $\alpha$.
  7. Let $\mathbb{R}^\omega$ be the set of sequences of real numbers, with the product topology. Let $\mathbb{R}^\infty$ be the set of all sequences of real numbers that are "eventually zero", ie., such that all but finitely many terms are zero. What is $\overline{\mathbb{R}^\infty}$?
  8. Prove that if $d$ is a metric on $X$ then the topology induced by $d$ is the coarsest topology such that $d \colon X \times X \to \mathbb{R}$ is continuous.

Due Oct 30

  1. Prove that $d(\vec{x},\vec{y}) = |x_1 - y_1| + \dots + |x_n - y_n|$ defines a metric on $\mathbb{R}^n$. Prove that it is equivalent to the usual metric.
  2. Do questions 4, 5, and 6 on page 127 of the edition I have of Munkres. Here is a scan.
  3. Recall that the lower limit topology on $\mathbb{R}$ has basis given by half-open intervals of the form $[a,b)$. Prove that it is first countable.
  4. Prove that if $p \colon X \to Y$ and $f \colon Y \to X$ are continuous maps such that $p \circ f$ is the identity on $Y$ then $p$ is a quotient map.
  5. Prove that a composition of two quotient maps is a quotient map.

Due Nov 17

  1. Prove that $(0,1)$ is not homeomorphic to $[0,1)$.
  2. Prove that $\mathbb{R}$ is not homeomorphic to $\mathbb{R}^2$.
  3. Prove that every continuous function $f \colon [0,1] \to [0,1]$ has a fixed point, that is, $x$ such that $f(x) = x$.
  4. Prove that $X \times X$ is path connected if and only if $X$ is.
  5. Suppose $A_\alpha$ are path connected subspaces of $X$, and $\bigcap A_\alpha \neq \emptyset$. Prove that $\bigcup A_\alpha$ is path connected.
  6. Prove that $\mathbb{R}^2 \setminus \mathbb{Q}^2$ is path connected.
  7. Prove that every connected open subset of $\mathbb{R}^2$ is path connected.

Due Dec 1

  1. Suppose $(X,\mathcal{T})$ is a compact Hausdorff space, and $(X,\mathcal{T}')$ is a different topology on the same set of points. Prove that if $\mathcal{T}'$ is finer than $\mathcal{T}$ then $(X,\mathcal{T}')$ is not compact, and if $\mathcal{T}'$ is coarser than $\mathcal{T}$ then $(X,\mathcal{T}')$ is not Hausdorff.
  2. Prove that if $A$ and $B$ are compact subsets of $X$ then so is $A \cup B$.
  3. Suppose $A$ is a subset of a metric space. Prove that if $A$ is compact then $A$ is closed and bounded. Give an example to show the converse does not hold.
  4. Prove that if $X$ is Hausdorff and $A$ and $B$ are disjoint compact subsets of $X$ then there exist disjoint open sets $U$ and $V$ containing $A$ and $B$ respectively.
  5. Prove that if $X$ is any topological space and $Y$ is a compact topological space then the projection map $X \times Y \to X$ is a closed map.
  6. Suppose $Y$ is a compact Hausdorff space, and $f \colon X \to Y$ is a not-necessarily-continuous function. Prove that $f$ is continuous if and only if its graph is a closed subset of the product space $X \times Y$.