Wondering how to use Paracompact in a sentence? Below are 10+ example sentences from authentic English texts. Including the meaning .
Paracompact meaning
In which every open cover admits an open locally finite refinement.
Using Paracompact
- The main meaning on this page is: In which every open cover admits an open locally finite refinement.
- In the example corpus, paracompact often appears in combinations such as: paracompact space, is paracompact, of paracompact.
Context around Paracompact
- Average sentence length in these examples: 16.2 words
- Position in the sentence: 5 start, 10 middle, 5 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis for Paracompact
- In this selection, "paracompact" usually appears in the middle of the sentence. The average example has 16.2 words, and this corpus slice is mostly made up of statements.
- Around the word, hereditarily, non, sometimes, space, spaces and hausdorff stand out and add context to how "paracompact" is used.
- Recognizable usage signals include space is paracompact and of a paracompact space and. That gives this page its own corpus information beyond isolated example sentences.
- By corpus frequency, "paracompact" sits close to words such as aare, aarti and abl, which helps place it inside the broader word index.
Example types with paracompact
The same corpus examples are grouped by length and sentence type, making it easier to see the contexts in which the word appears:
Every metric space is paracompact. (5 words)
Every compact space is Lindelöf and paracompact. (7 words)
In fact, for metric spaces, all subsets are paracompact. (9 words)
Tychonoff's theorem (which states that the product of any collection of compact topological spaces is compact) does not generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact. (34 words)
Proof that paracompact Hausdorff spaces admit partitions of unity A Hausdorff space is paracompact if and only if it every open cover admits a subordinate partition of unity. (28 words)
When it was proved by A.H. Stone that for Hausdorff spaces fully normal and paracompact are equivalent, he implicitly proved that all metrizable spaces are paracompact. (27 words)
Example sentences (20)
Although a product of paracompact spaces need not be paracompact, the following are true: * The product of a paracompact space and a compact space is paracompact.
Tychonoff's theorem (which states that the product of any collection of compact topological spaces is compact) does not generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact.
A space such that every subspace of it is a paracompact space is called hereditarily paracompact.
However, the product of a paracompact space and a compact space is always paracompact.
P ; Paracompact : A space is paracompact if every open cover has a locally finite open refinement.
Proof that paracompact Hausdorff spaces admit partitions of unity A Hausdorff space is paracompact if and only if it every open cover admits a subordinate partition of unity.
Properties Paracompactness is weakly hereditary, i.e. every closed subspace of a paracompact space is paracompact.
When it was proved by A.H. Stone that for Hausdorff spaces fully normal and paracompact are equivalent, he implicitly proved that all metrizable spaces are paracompact.
A topological space X is now said to be paracompact if every open cover has a locally finite open refinement.
Every compact space is Lindelöf and paracompact.
Every metric space is paracompact.
Every metrizable space is Hausdorff and paracompact (and hence normal and Tychonoff).
Every paracompact space is metacompact, and every metacompact space is orthocompact.
However, there exist non-paracompact manifolds which are not even normal.
In fact, for metric spaces, all subsets are paracompact.
It is different in these respects: * A paracompact subset of a Hausdorff space need not be closed.
One can check as an exercise that this provides an open refinement, since paracompact Hausdorff spaces are regular, and since is locally finite.
Sometimes paracompact spaces are defined so as to always be Hausdorff.
Steen Seebach (1978) p.23 Paracompact Hausdorff spaces are normal.
This is equivalent to requiring that every open subspace be paracompact.
Common combinations with paracompact
These word pairs occur most frequently in English texts:
- paracompact space 6×
- is paracompact 6×
- of paracompact 5×
- paracompact spaces 5×
- be paracompact 4×
- paracompact if 3×
- paracompact hausdorff 3×
- and paracompact 3×
- are paracompact 3×