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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.

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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.

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  • $\begingroup$ May I ask? So if X is a topology space with every convergent sequence is eventually constant, then X is T1? $\endgroup$ Jun 22, 2021 at 3:13
  • $\begingroup$ Yes, that's the contrapositive of the statement $\endgroup$
    – David Lui
    Jun 22, 2021 at 3:17
  • $\begingroup$ how do I guarantee that there are a and b such that every open set containing a and b ? $\endgroup$ Jun 22, 2021 at 3:22
  • $\begingroup$ @MartinFirdaus This is just the negation of the definition of a $T_1$ space. $\endgroup$ Jun 22, 2021 at 3:27
  • $\begingroup$ Let X={0,1} and \tau=\{X, empty set, {0}\}. A topological space (X, \tau) is not T1, but there is an open set {0} such that {0} doesn't contain 1. $\endgroup$ Jun 22, 2021 at 3:33

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