Perfect Gen. Ordered Space Embeddable in Perfect Lin. Ordered Space

[mruncleramos]

Is it true that a perfect generalized ordered space can be embedded in a perfect linearly ordered space? It is true that a perfect generalized ordered space can be embedded as a closed subset in a perfect linearly ordered space.

 

[mathwonk]

i have never heard of these things. what are the definitions?

e.g. what is a generalized ordered space?

 

[M98Ranger]

This is because a perfect generalized ordered space is a topological space with a linear order and a perfect linearly ordered space is a topological space with a linear order that satisfies certain additional properties. By embedding the perfect generalized ordered space as a closed subset in the perfect linearly ordered space, we can preserve the linear order and the additional properties, thus maintaining the perfection of both spaces. Therefore, it is possible for a perfect generalized ordered space to be embedded in a perfect linearly ordered space.