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Given an example of a space $X$ with property $P$, let $Y$ be the topological sum of $X$ with any countable non-metrizablesecond countable space. Then $Y$ has property $P$ but is not second-countable.
Given an example of a space $X$ with property $P$, let $Y$ be the topological sum of $X$ with any countable non-metrizable space. Then $Y$ has property $P$ but is not second-countable.
Given an example of a space $X$ with property $P$, let $Y$ be the topological sum of $X$ with any countable non-second countable space. Then $Y$ has property $P$ but is not second-countable.
Given an example of a space $X$ with property $P$, let $Y$ be the topological sum of $X$ with any countable non-metrizable space. Then $Y$ has property $P$ but is not second-countable.