Cardinality $\lt\mathfrak c$ or Smaller than the continuum ,Cardinality $\lt\beth_0$

The cardinality of the space is less than or equal to the cardinality of $\mathbb R$ .

See sections 1.12-1.15 of MR MR2048350.

Id	Spaces	Value	Source
1	Discrete topology on a two-point set
105	Helly space
132	Duncan's space

Cardinality <c\lt\mathfrak c<c or Smaller than the continuum ,Cardinality <ℶ0\lt\beth_0<ℶ0​