let $X$ denote the $\mathbb{R^2}$ endowed with the usual topology .let Y denote $\mathbb{R}$ endowed with cofinite topology . if Z is the product topology space $Y \times Y$,then
a) the topology of X is the same as the topology of Z
b) the topology of X is strictly coarser (weaker) than that of Z
c) the topology of Z is strictly coarser (weaker ) than that of X
d) the topology of X can not be compared with that of Z
My attempts : I thinks option option d) will correct as X is infinite and Y is finite
im just used my logics that infinite can not be compared with finite
Am i right or wrong
pliz verified
thanks u
Best Answer
Both $X$ and $Z$ are $\mathbb{R}^2$ as sets. Very infinite, so I don't get your remark.
$X$ is the product topology of $(\mathbb{R}, \mathcal{T}_e)$ with itself. $\mathcal{T}_e$ is the usual Euclidean topology on the reals.
It's clear that $\mathcal{T}_{\text{cof}} \subseteq \mathcal{T}_e$, as a metric topology is $T_1$, and strictly so.
So the topology on $Z$ is coarser (strictly) than that on $X$., so option c.