A note on D-property, monotone monolithicity and sigma-product

作者：Peng Liang Xue^*

来源：Topology and Its Applications, 2014, 161: 17-25.

DOI：10.1016/j.topol.2013.09.013

• D-space
• Monolithic
• Monotonically monolithic
• sigma-product
• C-p(X)

In this note we firstly show that if X is a a-product of Lindelof Sigma-spaces, then C-p(X) is a semi-monotonically monolithic space and hence it is monolithic.
In the second part of this note we show that if {X-alpha: alpha E is an element of A} is a family of topological spaces such that x, is a D-space for each n E N and alpha(1) is an element of Lambda for each i n, then any sigma-product of {X-n : alpha is an element of Lambda} is a D-space. This generalizes a. conclusion of A.V. Arhangel'skii. By this conclusion and a known conclusion we have that any sigma-product of regular weak (theta) over bar -refinable (or (sub)metacompact) C-scattered spaces is a D-space. We also show that if Y is a dense subspace of a topological space X with vertical bar X \ Y vertical bar <= omega such that C (Y) is monotonically monolithic, then C-p(X) is monotonically monolithic.
In the last part of this note we show that if X has a k-in-countable weak base for some k E N then X is a D-space.

• 出版日期2014-1-1
• 单位北京工业大学