Accumulation Points of Sequence of Distinct Terms in Infinite Particular Point Space/Mistake
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Source Work
1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.):
- Part $\text {II}$: Counterexamples
- Section $9 \text { - } 10$: Infinite Particular Point Topology
- Item $1$
- Section $9 \text { - } 10$: Infinite Particular Point Topology
Mistake
- The sequences $\set {a_i}$ which converge are those for which the $a_i \ne p$ are equal for all but a finite number of indices. The only accumulation points for sequences are the points $b_j$ that the $a_i$ equal for infinitely many indices.
Correction
It is essential to clarify that in the sentence:
- The only accumulation points for sequences are the points $b_j$ that the $a_i$ equal for infinitely many indices.
the sequences in question are specifically those where all $a_i \ne p$.
It is insufficient for complete understanding that the $a_i \ne p$ in the previous question (for which it is not even clear whether it should apply to all terms of $\set {a_i}$ or not) be an implicit condition on the sequences being referred to in the second sentence.
Also see
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $9 \text { - } 10$. Infinite Particular Point Topology: $1$