Fix a non-negative integer $d$. Let $$\mathcal{A}_d := \{A \subseteq \mathbb{C} : A \text{ is the zero-set of a polynomial of degree } \leq d \text{ in } \mathbb{C}[X]\}.$$ Let $\mathcal{T}$ be the coarsest topology on $\mathbb{C}$ in which $A$ is closed for every $A \in \mathcal{A}_d$. (A) Determine whether $\mathcal{T}$ is Hausdorff. (B) Show that for every polynomial $f(X) \in \mathbb{C}[X]$, the function $\mathbb{C} \longrightarrow \mathbb{C}$ defined by $z \mapsto f(z)$ is continuous, where $\mathbb{C}$ (on both the sides) is given the topology $\mathcal{T}$.
Fix a non-negative integer $d$. Let
$$\mathcal{A}_d := \{A \subseteq \mathbb{C} : A \text{ is the zero-set of a polynomial of degree } \leq d \text{ in } \mathbb{C}[X]\}.$$
Let $\mathcal{T}$ be the coarsest topology on $\mathbb{C}$ in which $A$ is closed for every $A \in \mathcal{A}_d$.\\
(A) Determine whether $\mathcal{T}$ is Hausdorff.\\
(B) Show that for every polynomial $f(X) \in \mathbb{C}[X]$, the function $\mathbb{C} \longrightarrow \mathbb{C}$ defined by $z \mapsto f(z)$ is continuous, where $\mathbb{C}$ (on both the sides) is given the topology $\mathcal{T}$.