Some topological properties of spaces between the Sorgenfrey and usual topologies on real number

Abstract: The

H

-space, denoted as

(m a t h b b R, a u_{A})

, has

m a t h b b R

as its point set and a basis consisting of usual open interval neighborhood at points of

A

while taking Sorgenfrey neighborhoods at points of

m a t h b b R

-

A

. In this paper, we mainly discuss some topological properties of

H

-spaces. In particular, we prove that, for any subset

A s u b s e t m a t h b b R

, (1)

(m a t h b b R, a u_{A})

is zero-dimensional iff

m a t h b b R s e t m i n u s A

is dense in

(m a t h b b R, a u_{E})

; (2)

(m a t h b b R, a u_{A})

is locally compact iff

(m a t h b b R, a u_{A})

is a

k_{o m e g a}

-space; (3) if

(m a t h b b R, a u_{A})

is

s i g m a

-compact, then

m a t h b b R s e t m i n u s A

is countable and nowhere dense; if

m a t h b b R s e t m i n u s A

is countable and scattered, then

(m a t h b b R, a u_{A})

is

s i g m a

-compact; (4)

(m a t h b b R, a u_{A})^{a l e p h_{0}}

is perfectly subparacompact; (5) there exists a subset

A s u b s e t m a t h b b R

such that

(m a t h b b R, a u_{A})

is not quasi-metrizable; (6)

(m a t h b b R, a u_{A})

is metrizable if and only if

(m a t h b b R, a u_{A})

is a

-space.

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