Variations
There are several variations of the notion of paracompactness. To define them, we first need to extend the list of terms above:
A topological space is:
- metacompact if every open cover has an open pointwise finite refinement.
- orthocompact if every open cover has an open refinement such that the intersection of all the open sets about any point in this refinement is open.
- fully normal if every open cover has an open star refinement, and fully T4 if it is fully normal and T1 (see separation axioms).
The adverb "countably" can be added to any of the adjectives "paracompact", "metacompact", and "fully normal" to make the requirement apply only to countable open covers.
Every paracompact space is metacompact, and every metacompact space is orthocompact.
Read more about this topic: Paracompact Space
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