Exercises Attempts

Exercise after §23

1. Let $\mathcal{T}$ and $\mathcal{T}'$ be two topologies on $X$. If $\mathcal{T}' \supseteq \mathcal{T}$, what does connectedness of $X$ in one topology imply about connectedness in the other?

2. Let $\{A_n\}$ be a sequence of connected subspaces of $X$, such that $A_n \cap A_{n+1} \neq \emptyset$ for all $n$. Show that $\bigcup A_n$ is connected.

3. Let $\{A_\alpha\}$ be a collection of connected subspaces of $X$; let $A$ be a connected subspace of $X$. Show that if $A \cap A_\alpha \neq \emptyset$ for all $\alpha$, then $A \cup (\bigcup A_\alpha)$ is connected.

4. Show that if $X$ is an infinite set, it is connected in the finite complement topology.

5. A space is totally disconnected if its only connected subspaces are one-point sets. Show that if $X$ has the discrete topology, then $X$ is totally disconnected. Does the converse hold?

9. Let $A$ be a proper subset of $X$, and let $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, show that

$$(X \times Y) \backslash (A \times B)$$

is connected.

10. Let $\{X_\alpha\}_{\alpha \in J}$ be an indexed family of connected spaces; let $X$ be the product space

    $$X=\prod_{\alpha \in J} X_\alpha.$$

    Let $\mathbf{a}=(a_\alpha)$ be a fixed point of $X$.

    (a) Given any finite subset $K$ of $J$, let $X_K$ denote the subspace of $X$ consisting of all points $\mathbf{x}=(x_\alpha)$ such that $x_\alpha=a_\alpha$ for $\alpha \in K$. Show that $X_K$ is connected.

    (b) Show that the union $Y$ of the spaces $X_K$ is connected.

    (c) Show that $X$ equals the closure of $Y$; conclude that $X$ is connected.

12. Let $Y \subseteq X$; let $X$ and $Y$ be connected. Show that if $A$ and $B$ form a separation of $X \backslash Y$, then $Y \cup A$ and $Y \cup B$ are connected.

Exercise after §24

1. (a) Show that no two of the spaces $(0,1)$, $(0,1]$, and $[0,1]$ are homeomorphic. [Hint: What happens if you remove a point from each of these spaces?]

   (b) Suppose that there exists embeddings $f : X \to Y$ and $g : Y \to X$. Show by means of an example that $X$ and $Y$ need not be homeomorphic.

   (c) Show $\mathbb{R}^n$ and $\mathbb{R}$ are not homeomorphic if $n>1$.

2. Let $f : S^1 \to \mathbb{R}$ be a continuous map. Show there exists a point $x$ of $S^1$ such that $f(x)=f(-x)$.

note

Recall that $S^1$ is defined as the space of all complex numbers $z$ for which $|z|=1$.

3. Let $f : X \to X$ be continuous. Show that if $X = [0,1]$, there is a point $x$ such that $f(x)=x$. The point $x$ is called a fixed point of $f$. What happens if $X$ equals $[0,1)$ or $(0,1)$?

4. Let $X$ be an ordered set in the order topology. Show that if $X$ is connected, then $X$ is a linear continuum.

10. Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected. [Hint: Show that given $x_0 \in U$, the set of points that can be joined to $x_0$ by a path in $U$ is both open and closed in $U$.]