Class of extensions of real field and their topological properties

Key Points

• Extensions of real field may be connected or not, impacting their closure under binary operations.
• Connected extensions are not closed under addition and multiplication, whereas non-connected ones form a linearly ordered field.
• These findings suggest possible applications in constructing measures for sets of zero Lebesgue measure.
• A deeper understanding of these properties could enhance the study of real fields and their extensions in topology.