All Questions page - 62

Let $X$ be a topological space and $Y$ be a topological space with ordered topology. Let $f,g:X \rightarrow Y$ be continuous functions. Show that the set $U:=\{x \in X:f(x)>g(x) \}$ is open in $X$.

Prove: Composition of continuous functions is continuous.

Let $X:=[0,1]$ and $A_0:=\{0\} , A_n:=[1/n,1]$. Then the function $f:[0,1] \rightarrow \mathbb{R}$given by $f(0)=1,f(x)=1/x(0<x\le1)$ is discontinuous at 0. However, $f|_{A_n}$ is continuous for every $n \ge 0$.

Show that the function $g: (0,1) \rightarrow \mathbb{R}$ given below is continuous on irrationals and discontinuous on rationals:

$g(x) = \frac{1}{q} \; if \; x \in \mathbb{Q} \; \cap \; (0,1)$and $x = \frac{p}{q}$ in reduced

form = 0 otherwise.

We say that a set is a $G_{\delta}$ set if it is countable intersection of open sets.

Show that a bijective continuous map from a compact metric space into a metric space sends closed sets to closed sets, and hence it is a homeomorphism.

Define $f: \mathbb{R} \rightarrow [0,1]$ by $f(x) =x$ if $|x| \le 1,$ and $f(x)= \frac{1}{|x|}$ if $|x| \ge1$. Then f is continuous on $\mathbb{R}$. Note that $f$ is onto but not one-one.

 Show that the interval $(a,b) \subseteq \mathbb{R}$ is homeomorphic to any other interval $(c,d) \subseteq \mathbb{R}$.

Show that $e^{-x}$ is a homeomorphism from $(0, \infty)$ onto $(0,1)$.

Which of spaces $X$ and $Y$ are homeomorphic:

(1) $X= \mathbb{R}$ and $Y=[0,1)$

(2) $X=\mathbb{R}$ and $Y=[0,1]$

(3) $X=[1, \infty)$ and $Y=(0,1]$

(4) $X=\mathbb{R}$ and $Y=(0,1)$

(5) $X=\mathbb{Q}$ and $Y=\mathbb{Q}$

(6) $X=\{(x,y,z) \in \mathbb{R^3} : x^2+y^2 =z ,r<z<R \}$ and $Y =A(0,r,R)$

(7) $X= \{x \in \mathbb{R^{n+1}}:x_{n+1}=0 \}$ and $Y= \mathbb{R^n}$

Verify that $\phi(t)=e^{it}$ is a homeomorphism between $(0,2\pi)$ and $\mathbb{T} \backslash \{1\}$, where $\mathbb{T}$ denotes the unit circle in the complex plane. Conclude that unit circle minus a point is homeomorphic to the real line.

Show that $X=A(0,r,R)$ (open annulus) and $Y= \mathbb{T}$(unit circle) are not homeomorphic

Let $X=S^n \backslash \{(0,\ldots ,1)\} \subseteq \mathbb{R^{n+1}}$ and $Y= \mathbb{R^n}$ , and define

$f:X \rightarrow Y$ and $g:Y \rightarrow X$ by

$f(x_1, \ldots ,x_{n+1}) =\large( \frac{x_1}{1-x_{n+1}}, \ldots , \frac{x_n}{1-x_{n+1}}),\\g(x_1, \ldots ,x_n)=\large(\frac{2x_1}{{||x||_2}^2+1} ,\ldots ,\frac{2x_n}{{||x||_2}^2+1} , \frac{{||x||_2}^2-1}{{||x||_2}^2+1} )$

Verify that $f$ and $g$ are homeomorphisms such that $f^{-1}=g$.

Show that the graph $G$ of a continuous function $f:X \rightarrow Y$ (with the subspace topology inherited from $X \times Y$ ) is homeomorphic to $X$.

Consider the continuous function $f: \mathbb{R} \backslash \{0\} \rightarrow \mathbb{R}$ by $f(x)=1/x$. The graph of f is the hyperbola $xy=1$. One may conclude from the last exercise that $\mathbb{R} \backslash \{0\}$ is homeomorphic to $xy=1$.

Consider the metric spaces $(X_i,d_i),i=1 ,\cdots ,m$. Check that $d(x,y)=max_id_i(x_i,y_i)$ defined on $X= {\Pi ^m}_{i=1} X_i$ is a metric. The metric topology coincides with the product topology on $X$. This may be concluded from the fact that for $x=(x_1, \cdots ,x_m),\mathbb{B_d}(x,r) ={\Pi ^m}_{i=1} \mathbb{B_{d_i}}(x_i,r)$.

Show that projections $\pi_j:{\Pi^m}_{i=1}X_i \rightarrow X_j$ is continuous.

Verify that $\mathbb{B_b}$ and $\mathbb{B_p}$ are basis for a topology on ${\Pi^m}_\alpha X_\alpha$:

(1)  $\mathbb{B_b}=\{\Pi_{\alpha \in I } U_\alpha :U_\alpha \; is \; open \; in X_\alpha \}$ .

(2) $\mathbb{B_b}=\{\Pi_{\alpha \in I } U_\alpha \in \mathbb{B_b}:U_\alpha=X_\alpha$ for finitely many values of $\alpha\}$.

Suppose that each $X_\alpha$ contains a non-empty proper open subset $U_\alpha$. Show that $\Omega_\mathbb{B_p} = \Omega_\mathbb{B_b}$ iff $I$ has finite cardinality.