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prabau authored Oct 7, 2024
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A space in which every open cover has a $\theta$-sequence of open refinements; that is, a space $X$ in which, for every open cover $\mathscr U$, there exists a sequence $\langle \mathscr V_n : n \in \omega\rangle$ of open covers where each $\mathscr V_n$ is a refinement of $\mathscr U$ and, for each point $x \in X$, there exists $n \in \omega$ such that $\mathscr V_n$ is point-finite at $x$.

This property was introduced in {{zb:0132.18401}} under the name of <i>$\theta$-refinable</i>, and later renamed to <i>submetacompactness</i> in {{zb:0413.54027}} (full article available [here](http://topology.nipissingu.ca/tp/reprints/v03/tp03207s.pdf)).
This property was introduced in {{zb:0132.18401}} under the name of *$\theta$-refinable*, and later renamed to *submetacompact* in {{zb:0413.54027}} (full article available [here](http://topology.nipissingu.ca/tp/reprints/v03/tp03207s.pdf)).

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