The map $D$ is choose the correct satements

Question

Consider the closed interval $[0, 1]$ in the real line $\mathbb{R}$ and the product space $([0, 1]^{\mathbb{N}}, τ ),$

where $τ$ is a topology on $[0, 1]^\mathbb{N} $. Let $ D : [0, 1] \rightarrow [0, 1]^\mathbb{N} $ be the map defined by $D(x) := (x, x, · · · , x, · · ·)$ for $x \in [0, 1].$ . The map $D$ is

choose the correct satements

$(a)$ not continuous if $τ$ is the box topology and also not continuous if $τ$ is the product topology.

$(b)$ continuous if $τ$ is the product topology and also continuous if $τ$ is the box topology.

$(c)$ continuous if $τ$ is the box topology and not continuous if $τ$ is the product topology.

$(d)$ continuous if $τ$ is the product topology and not continuous if $τ$ is the box topology

My attempt : i thinks option $c)$ is true because box topology is finer then product

Is its true ?

Any hints/solution

Answer 1

The correct answer is d). Since $D^{-1}\prod (-\frac 1 n,\frac 1 n)=\{0\}$ is not open it follows that $D$ is not continuous for the box topology. If $x_j \to x$ then $D(x_j)\to D(x)$ in the product topolgy because convergence in product topolgy is coordinatewise convergence. Hence $D$ is continuous for product topology.

Answer 2

Hint: what is the inverse image by $D$ of open subsets from $\tau$ when $\tau$ is the box topology, or the product topology?