# Non T1 space with every convergent sequence is eventually constant

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.

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.

• May I ask? So if X is a topology space with every convergent sequence is eventually constant, then X is T1? Jun 22 at 3:13
• Yes, that's the contrapositive of the statement Jun 22 at 3:17
• how do I guarantee that there are a and b such that every open set containing a and b ? Jun 22 at 3:22
• @MartinFirdaus This is just the negation of the definition of a $T_1$ space. Jun 22 at 3:27
• 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. Jun 22 at 3:33