Definition:Rational Sequence Topology
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Definition
Let $\struct {\R, \tau_d}$ denote the real number line with the usual (Euclidean) topology.
Let $\Bbb I := \R \setminus \Q$ denote the set of irrational numbers
For each $x \in \Bbb I$, let $\sequence {x_i}$ be a sequence of rational numbers which converges to $x$ in $\tau_d$.
Let $\tau$ be the topology defined on $\R$ as:
- $(1): \quad$ All rational numbers are open points in $\R$
- $(2): \quad$ The sets $U_n$ of the form:
- $\map {U_n} x := \sequence {x_i}_n^\infty \cup \set x$
- form a basis for the irrational point $x$.
$\tau$ is then referred to as the rational sequence topology on $\R$.
This page needs proofreading. In particular: This went somewhat over my head. Anyone care to try and make it a bit more understandable? The concept of the "basis for the irrational point $x$" I don't get. If you believe all issues are dealt with, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |
Also see
- Results about rational sequence topology can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (next): Part $\text {II}$: Counterexamples: $65$. Rational Sequence Topology