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I think the Zariski topology from a subfield provides even more natural examples. You can define a Zariski topology from Q on $C^n$ so that the closed sets are zero sets of polynomials with rational coefficients. Then

1) Because $\pi$ is transcendental, the closure of ($\pi$,0) in $C^2$ in this topology is the x-axis y=0. (This topology is not $T_1$.) The constant sequence such that every point is ($\pi$,0) converges to every point on the x-axis.

2) If $\alpha$ and $\beta$ are algebraically independent transcendentals, then the constant sequence {($\alpha$,$\beta$)} converges to every point.

Another natural non-Hausdorff space , is the quotient topology on leaves of a foliation. Consider the foliation of $R^2$ by vertical lines y=a x=a for a≤-1 or a≥1, and by parallel U-shaped leaves, y=$1/(1-x^2)+C$ where -1<x<1. Then a sequence of leaves with $C$ -> -$\infty$ converges both to the leaf x=-1 and the leaf x=+1.
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I think the Zariski topology from a subfield provides even more natural examples. You can define a Zariski topology from Q on $C^n$ so that the closed sets are zero sets of polynomials with rational coefficients. Then

1) Because $\pi$ is transcendental, the closure of ($\pi$,0) in $C^2$ in this topology is the x-axis y=0. (This topology is not $T_1$.) The constant sequence such that every point is ($\pi$,0) converges to every point on the x-axis.

2) If $\alpha$ and $\beta$ are algebraically independent transcendentals, then the constant sequence {($\alpha$,$\beta$)} converges to every point.

Another natural non-Hausdorff space, is the quotient topology on leaves of a foliation. Consider the foliation of $R^2$ by vertical lines y=a for a≤-1 or a≥1, and by parallel U-shaped leaves, y=$1/(1-x^2)+C$ where -1<x<1. Then the a sequence of leaves with $C$ -> -$\infty$ converges both to the leaf x=-1 and the leaf x=+1.
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I think the Zariski topology from a subfield provides even more natural examples. You can define a Zariski topology from Q on $C^n$ so that the closed sets are zero sets of polynomials with rational coefficients. Then

1) Because $\pi$ is transcendental, the closure of ($\pi$,0) in $C^2$ in this topology is the x-axis y=0. (This topology is not $T_1$.) The constant sequence such that every point is ($\pi$,0) converges to every point on the x-axis.

2) If $\alpha$ and $\beta$ are algebraically independent transcendentals, then the constant sequence {($\alpha$,$\beta$)} converges to every point.

Another natural non-Hausdorff space, is the quotient topology on leaves of a foliation. Consider the foliation of $R^2$ by vertical lines y=a for a≤-1 or a≥1, and by parallel U-shaped leaves, y=$1/(1-x^2)+C$ where -1<x<1. Then the sequence of leaves with $C$ -> -$\infty$ converges both to the leaf x=-1 and the leaf x=+1.