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Let X=R$^{2}$ with cofinite topology and consider p$_{n}$(z)=z$^{2}$+$\frac{1}{n^{2}(n^{2}-1)}$z-$\frac{1}{n^{2}(n^{2}-1)}$ be the sequence of polynomials. Define B={b$_{n}$$|$ p$_{n}$(b$_{n}$)=0 n$\in\mathbb{N}$} be a subset of complex numbers regarded as a sequence in X. What are the limit points of B?

I also have this question to refer to (Which sequences converge in a cofinite topology and what is their limit?) which says that given any sequence (x$_{n}$) in cofinite topology: a) If there exists no value which the sequence takes infinitely many times, then sequence converges to every x$\in$X.

b) If there exists exactly one value which the sequence takes infinitely many times, then only that value is the limit.

c) If there exist two or more values which the sequence takes infinitely many times, then sequence diverges.

Now, I am confused as to how to approach my original question.

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HumbleStudent
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1 Answers1

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Your question is about finding limit points of the set $B=\{b_n|p_n(b_n)=0\} $. Any polynomial $p_n$ has two roots, and what you need do is to determine the set $B$ is of which type a), b) or c) mentioned above. Note that your question is about the behavior of $B$, not the cofinite topology.

Qurultay
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