Analysis And Topology

# Analysis And Topology

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Paper 1, Section II, F

commentLet $f: X \rightarrow Y$ be a map between metric spaces. Prove that the following two statements are equivalent:

(i) $f^{-1}(A) \subset X$ is open whenever $A \subset Y$ is open.

(ii) $f\left(x_{n}\right) \rightarrow f(a)$ for any sequence $x_{n} \rightarrow a$.

For $f: X \rightarrow Y$ as above, determine which of the following statements are always true and which may be false, giving a proof or a counterexample as appropriate.

(a) If $X$ is compact and $f$ is continuous, then $f$ is uniformly continuous.

(b) If $X$ is compact and $f$ is continuous, then $Y$ is compact.

(c) If $X$ is connected, $f$ is continuous and $f(X)$ is dense in $Y$, then $Y$ is connected.

(d) If the set $\{(x, y) \in X \times Y: y=f(x)\}$ is closed in $X \times Y$ and $Y$ is compact, then $f$ is continuous.

Paper 2, Section I, $2 F$

commentLet $K:[0,1] \times[0,1] \rightarrow \mathbb{R}$ be a continuous function and let $C([0,1])$ denote the set of continuous real-valued functions on $[0,1]$. Given $f \in C([0,1])$, define the function $T f$ by the expression

$T f(x)=\int_{0}^{1} K(x, y) f(y) d y$

(a) Prove that $T$ is a continuous map $C([0,1]) \rightarrow C([0,1])$ with the uniform metric on $C([0,1])$.

(b) Let $d_{1}$ be the metric on $C([0,1])$ given by

$d_{1}(f, g)=\int_{0}^{1}|f(x)-g(x)| d x$

Is $T$ continuous with respect to $d_{1} ?$

Paper 2, Section II, F

commentLet $k_{n}: \mathbb{R} \rightarrow \mathbb{R}$ be a sequence of functions satisfying the following properties:

$k_{n}(x) \geqslant 0$ for all $n$ and $x \in \mathbb{R}$ and there is $R>0$ such that $k_{n}$ vanishes outside $[-R, R]$ for all $n$

each $k_{n}$ is continuous and

$\int_{-\infty}^{\infty} k_{n}(t) d t=1$

- given $\varepsilon>0$ and $\delta>0$, there exists a positive integer $N$ such that if $n \geqslant N$, then

$\int_{-\infty}^{-\delta} k_{n}(t) d t+\int_{\delta}^{\infty} k_{n}(t) d t<\varepsilon$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a bounded continuous function and set

$f_{n}(x):=\int_{-\infty}^{\infty} k_{n}(t) f(x-t) d t$

Show that $f_{n}$ converges uniformly to $f$ on any compact subset of $\mathbb{R}$.

Let $g:[0,1] \rightarrow \mathbb{R}$ be a continuous function with $g(0)=g(1)=0$. Show that there is a sequence of polynomials $p_{n}$ such that $p_{n}$ converges uniformly to $g$ on $[0,1]$. $[$ Hint: consider the functions

$k_{n}(t)= \begin{cases}\left(1-t^{2}\right)^{n} / c_{n} & t \in[-1,1] \\ 0 & \text { otherwise }\end{cases}$

where $c_{n}$ is a suitably chosen constant.]

Paper 3, Section II, F

commentDefine the terms connected and path-connected for a topological space. Prove that the interval $[0,1]$ is connected and that if a topological space is path-connected, then it is connected.

Let $X$ be an open subset of Euclidean space $\mathbb{R}^{n}$. Show that $X$ is connected if and only if $X$ is path-connected.

Let $X$ be a topological space with the property that every point has a neighbourhood homeomorphic to an open set in $\mathbb{R}^{n}$. Assume $X$ is connected; must $X$ be also pathconnected? Briefly justify your answer.

Consider the following subsets of $\mathbb{R}^{2}$ :

$\begin{gathered} A=\{(x, 0): x \in(0,1]\}, \quad B=\{(0, y): y \in[1 / 2,1]\}, \text { and } \\ C_{n}=\{(1 / n, y): y \in[0,1]\} \text { for } n \geqslant 1 \end{gathered}$

Let

$X=A \cup B \cup \bigcup_{n \geqslant 1} C_{n}$

with the subspace topology. Is $X$ path-connected? Is $X$ connected? Justify your answers.

Paper 4, Section I, $2 F$

commentLet $X$ be a topological space with an equivalence relation, $\tilde{X}$ the set of equivalence classes, $\pi: X \rightarrow \tilde{X}$, the quotient map taking a point in $X$ to its equivalence class.

(a) Define the quotient topology on $\tilde{X}$ and check it is a topology.

(b) Prove that if $Y$ is a topological space, a map $f: \tilde{X} \rightarrow Y$ is continuous if and only if $f \circ \pi$ is continuous.

(c) If $X$ is Hausdorff, is it true that $\tilde{X}$ is also Hausdorff? Justify your answer.

Paper 4, Section II, F

comment(a) Let $g:[0,1] \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ be a continuous function such that for each $t \in[0,1]$, the partial derivatives $D_{i} g(t, x)(i=1, \ldots, n)$ of $x \mapsto g(t, x)$ exist and are continuous on $[0,1] \times \mathbb{R}^{n}$. Define $G: \mathbb{R}^{n} \rightarrow \mathbb{R}$ by

$G(x)=\int_{0}^{1} g(t, x) d t$

Show that $G$ has continuous partial derivatives $D_{i} G$ given by

$D_{i} G(x)=\int_{0}^{1} D_{i} g(t, x) d t$

for $i=1, \ldots, n$.

(b) Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be an infinitely differentiable function, that is, partial derivatives $D_{i_{1}} D_{i_{2}} \cdots D_{i_{k}} f$ exist and are continuous for all $k \in \mathbb{N}$ and $i_{1}, \ldots, i_{k} \in\{1,2\}$. Show that for any $\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}$,

$f\left(x_{1}, x_{2}\right)=f\left(x_{1}, 0\right)+x_{2} D_{2} f\left(x_{1}, 0\right)+x_{2}^{2} h\left(x_{1}, x_{2}\right)$

where $h: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is an infinitely differentiable function.

[Hint: You may use the fact that if $u: \mathbb{R} \rightarrow \mathbb{R}$ is infinitely differentiable, then

$\left.u(1)=u(0)+u^{\prime}(0)+\int_{0}^{1}(1-t) u^{\prime \prime}(t) d t .\right]$

Paper 1, Section II, E

commentState what it means for a function $f: \mathbb{R}^{m} \rightarrow \mathbb{R}^{r}$ to be differentiable at a point $x \in \mathbb{R}^{m}$, and define its derivative $f^{\prime}(x) .$

Let $\mathcal{M}_{n}$ be the vector space of $n \times n$ real-valued matrices, and let $p: \mathcal{M}_{n} \rightarrow \mathcal{M}_{n}$ be given by $p(A)=A^{3}-3 A-I$. Show that $p$ is differentiable at any $A \in \mathcal{M}_{n}$, and calculate its derivative.

State the inverse function theorem for a function $f$. In the case when $f(0)=0$ and $f^{\prime}(0)=I$, prove the existence of a continuous local inverse function in a neighbourhood of 0 . [The rest of the proof of the inverse function theorem is not expected.]

Show that there exists a positive $\epsilon$ such that there is a continuously differentiable function $q: D_{\epsilon}(I) \rightarrow \mathcal{M}_{n}$ such that $p \circ q=\left.\mathrm{id}\right|_{D_{\epsilon}(I)}$. Is it possible to find a continuously differentiable inverse to $p$ on the whole of $\mathcal{M}_{n}$ ? Justify your answer.

Paper 2, Section I, $2 E$

commentLet $\tau$ be the collection of subsets of $\mathbb{C}$ of the form $\mathbb{C} \backslash f^{-1}(0)$, where $f$ is an arbitrary complex polynomial. Show that $\tau$ is a topology on $\mathbb{C}$.

Given topological spaces $X$ and $Y$, define the product topology on $X \times Y$. Equip $\mathbb{C}^{2}$ with the topology given by the product of $(\mathbb{C}, \tau)$ with itself. Let $g$ be an arbitrary two-variable complex polynomial. Is the subset $\mathbb{C}^{2} \backslash g^{-1}(0)$ always open in this topology? Justify your answer.

Paper 2, Section II, E

commentLet $C[0,1]$ be the space of continuous real-valued functions on $[0,1]$, and let $d_{1}, d_{\infty}$ be the metrics on it given by

$d_{1}(f, g)=\int_{0}^{1}|f(x)-g(x)| d x \quad \text { and } \quad d_{\infty}(f, g)=\max _{x \in[0,1]}|f(x)-g(x)|$

Show that id : $\left(C[0,1], d_{\infty}\right) \rightarrow\left(C[0,1], d_{1}\right)$ is a continuous map. Do $d_{1}$ and $d_{\infty}$ induce the same topology on $C[0,1]$ ? Justify your answer.

Let $d$ denote for any $m \in \mathbb{N}$ the uniform metric on $\mathbb{R}^{m}: d\left(\left(x_{i}\right),\left(y_{i}\right)\right)=\max _{i}\left|x_{i}-y_{i}\right|$. Let $\mathcal{P}_{n} \subset C[0,1]$ be the subspace of real polynomials of degree at most $n$. Define a Lipschitz map between two metric spaces, and show that evaluation at a point gives a Lipschitz map $\left(C[0,1], d_{\infty}\right) \rightarrow(\mathbb{R}, d)$. Hence or otherwise find a bijection from $\left(\mathcal{P}_{n}, d_{\infty}\right)$ to $\left(\mathbb{R}^{n+1}, d\right)$ which is Lipschitz and has a Lipschitz inverse.

Let $\tilde{\mathcal{P}}_{n} \subset \mathcal{P}_{n}$ be the subset of polynomials with values in the range $[-1,1]$.

(i) Show that $\left(\tilde{\mathcal{P}}_{n}, d_{\infty}\right)$ is compact.

(ii) Show that $d_{1}$ and $d_{\infty}$ induce the same topology on $\tilde{\mathcal{P}}_{n}$.

Any theorems that you use should be clearly stated.

[You may use the fact that for distinct constants $a_{i}$, the following matrix is invertible:

$\left(\begin{array}{ccccc} 1 & a_{0} & a_{0}^{2} & \ldots & a_{0}^{n} \\ 1 & a_{1} & a_{1}^{2} & \ldots & a_{1}^{n} \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & a_{n} & a_{n}^{2} & \ldots & a_{n}^{n} \end{array}\right)$