I know that discrete space and cocountable space are $T1$ space with every convergent sequence is eventually constant (If we have a convergent sequence $(x_n),$ then $\exists\, n_0 \in N, \forall n\geq n_0, x_{n_0}=x_n ). $ Is there any example of a non T1 space with every convergent sequence is eventually constant? Thank you in advance.
2026-03-29 06:54:57.1774767297
Non T1 space with every convergent sequence is eventually constant
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Let $X$ be a non-T1 space, and let $a, b$ be two points such that every open set containing $a$ also contains $b$. Then, the sequence $\{a , b, a, b, ...\}$ is contained in every open set containing $a$, but it's not constant.