Which one of the following are true statements?

Question

Which one of the following are true? Let $X=(0,1)$ and $\mathscr T=\{\emptyset,X\}\cup \{(1/n,1):n\in \mathbb N\}$. Then $(X,\mathscr T)$ are:

(A)$T_1 $

(B)$T_2 $

(C)Regular

(D)Normal

(E) None of the above.

My solution:-

(A) Let $.67$ and $.70$ are in $(0,1)$. We can not find an open sets $U$ and $V$ in $(X,\mathscr T):$ such that $.67\in U$ and $.70\notin V$. So, $(X,\mathscr T)$ is not $T_1$.

(B)Let $.67$ and $.70$ are in $(0,1)$. We can not find disjoint open sets $U$ and $V$ in $(X,\mathscr T) $. Open sets containg $.67$ also contains $.70$. So, $(X,\mathscr T) $ is not $T_2$.

(C) consider a closed set $(0,\frac{1}{n_1}]$ and $\frac{1}{n_1}<x<1$. We can not find two disjoint open sets such that one contains $(0,\frac{1}{n_1}]$ and other contain $x$.

(D)We can not find two disjoint closed subsets of $X$. So, $(X,\mathscr T)$ is trivially Normal.

So, $(D)$ is the only correct answer. Am I correct?

Answer 1

Your proof regarding $T_1$ is fine. I just wanted to point out that there's a more clever way to reach the same result. This particular topology has only countably many open sets, but the interval $(0, 1)$ is uncountable. So there must be a pair of points that can't be separated by any open set in the topology.