Category:Rational Sequence Topology
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This category contains results about Rational Sequence Topology.
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 $\R$.
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