It seems all conditions in definition of a stratifiable space hold. But the product of 2 copy of Sorgenfrey lines is not normal, countably many products of stratifiable spaces is stratifiable. One has always Stratifiable =>Normal
A T1 space X is stratifiable if for each open set U of X one can assign a sequence {Un} (n E N) of open sets such that following conditions hold:
1. the closure of each Un is a subset of U;
2. Union of all sets Un is equal to U.
3. Un is a subset of Vn, whenever U is a subset of V.
The Sorgenfrey line not stratifiable, since (a, b] is open but condition 2 in the above definition not satisfied.
The Sorgenfrey line not stratifiable, since (a, b] is open but condition 2 in the above definition not satisfied.
(a,b] is closed but not open. [a,b) is open in the Sorgenfrey line. Get open sets [a, b-1/n). Condition 2 holds.
In terms of separation axioms, {\displaystyle \mathbb {R} _{l}} is a perfectly normal Hausdorff space.
In terms of countability axioms, {\displaystyle \mathbb {R} _{l}} is first-countable and separable, but not second-countable.
In terms of compactness properties, {\displaystyle \mathbb {R} _{l}} is Lindelöf and paracompact, but not σ-compact nor locally compact.
{\displaystyle \mathbb {R} _{l}} is not metrizable, since separable metric spaces are second-countable. However, the topology of a Sorgenfrey line is generated by a premetric.
I think it is related to 2nd countability
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