问题 1. Consider the following cases of a set $X$ and a collection $\tau \subseteq P(X)$ of subsets of $X$ :
(a) $\quad X=\{1,2,3,4\}, \quad \tau=\{\emptyset, X,\{1\},\{1,3\},\{2,3,4\}\}$
(b) $\quad X=\{1,2,3,4,5,6\}, \quad \tau=\{\emptyset, X,\{3,4\},\{1,2,3,4\},\{3,4,5,6\}\}$
(c) $\quad X=\mathbb{R}, \quad \tau=\{\mathbb{R}, A \subseteq \mathbb{R} \mid \mathbb{R} \backslash A$ an uncountable set $\}$
In each of these cases, determine which of the three properties (O1), (O2) and (O3) of a topology are satisfied by $\tau$. Justify your answers. Hence determine whether or not $\tau$ defines a topology on the given set $X$. [12 marks]

问题 2.

(a) Let $(X, \tau)$ be a topological space. Let $A \subseteq X .$ Let $x \in A$.
Let $\mathfrak{B}(x)$ be a neighbourhood basis of $x$ in the topology $\tau$ on $X$.
Prove that
$$
\{B \cap A \mid B \in \mathfrak{B}(x)\}
$$
is a neighbourhood basis of $x$ in the induced topology $\left.\tau\right|_{A}$ on $A$. marks]
(b) Let $X=\mathbb{R}$. For each $x \in \mathbb{R}$, let
$$
\mathfrak{B}(x):=\{(y, x] \mid y \in \mathbb{R}, y<x\}
$$
be a neighbourhood basis of $x$ which defines the topology $\tau$ on $\mathbb{R}$. Let $A=[0,1] \subseteq \mathbb{R}$. Use the result in (a) to find a neighbourhood basis of a point $x \in A$ in the induced topology $\left.\tau\right|_{A}$ on $A$.
$\{$ Hint: Consider the cases $x=0$ and $0<x \leq 1$ separately, and simplify your results. $\}$ $[5$ marks $]$

问题 3.

Consider the topological space $(\mathbb{R}, \sigma)$ with standard metric topology $\sigma$ on $\mathbb{R}$. Prove that the topological space $(\mathbb{R}, \sigma)$ is second countable. $\quad$ [6 marks]

问题 4.

Let $(X, \tau)$ be a topological space. Let $A \subseteq X$ and $B \subseteq X$. Use the definition of the interior of a subset to prove that
$$
(A \cup B)^{\circ} \supseteq A^{\circ} \cup B^{\circ}
$$
Determine whether or not
$$
(A \cup B)^{\circ} \subseteq A^{\circ} \cup B^{\circ}
$$
If so, give a proof. If not, give a counterexample.

问题 5.

Let $X=\mathbb{R}^{2}$ with standard metric topology $\tau$ on $\mathbb{R}^{2},$ and let $Y=\mathbb{R}$ with standard metric topology $\sigma$ on $\mathbb{R}$. Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be the map
$$
f(x, y):=x+y \quad \text { for } \quad(x, y) \in \mathbb{R}^{2}
$$
Prove that the function $f$ is continuous. Determine whether or not $f$ is a homeomorphism. Justify your answer. $\quad$ [12 marks]