# 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 $\R$**.

## 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