Category:Definitions/Pointed Extensions of Reals

This category contains definitions related to Pointed Extensions of Reals.
Related results can be found in Category:Pointed 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 defined as:

$\BB = \set {\set x \cup \paren {U \cap D}: x \in U \in \tau_d}$

Let $\tau'$ be the topology generated from $\BB$.

$\tau'$ is referred to as a pointed extension of $\R$.