Category:Indiscrete Extensions of Reals

This category contains results about Indiscrete Extensions of Reals.
Definitions specific to this category can be found in Definitions/Indiscrete Extensions of Reals.

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $D$ be an everywhere dense subset of $\struct {\R, \tau_d}$ with an everywhere dense complement in $\R$.

Let $\BB$ be the set of sets:

$\BB := \set {H: \exists U \in \tau_d: H = U \cap D}$

Let $\tau^*$ be the topology generated from $\tau_d$ by the addition of all sets of $\BB$.

$\tau^* = \tau_d \cup \BB$

$\tau^*$ is then referred to as an indiscrete extension of $\R$.