Below you will find example sentences with "hausdorff spaces". The examples show how this phrase is used in natural context and which words often surround it.
Hausdorff Spaces in a sentence
Corpus data
- Displayed example sentences: 20
- Discovered as a combination around: spaces
- Corpus frequency in the collocation scan: 9
- Phrase length: 2 words
- Average sentence length: 23 words
Sentence profile
- Phrase position: 4 start, 11 middle, 5 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis
- The phrase "hausdorff spaces" has 2 words and usually appears in the middle in these examples. The average sentence has 23 words and is mostly made up of statements.
- Around this phrase, patterns and context words such as locally compact hausdorff spaces, 23 paracompact hausdorff spaces are normal, compact, space and regular stand out.
- In the phrase index, this combination connects with parking spaces, vector spaces, topological spaces, vector spaces, topological spaces and public spaces, linking the page to nearby combinations.
Example types with hausdorff spaces
This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:
Embeddings into compact Hausdorff spaces may be of particular interest. (10 words)
Steen Seebach (1978) p.23 Paracompact Hausdorff spaces are normal. (10 words)
In fact, for Hausdorff spaces, paracompactness and full normality are equivalent. (11 words)
However, the compact Hausdorff spaces are " absolutely closed ", in the sense that, if you embed a compact Hausdorff space K in an arbitrary Hausdorff space X, then K will always be a closed subset of X; the "surrounding space" does not matter here. (43 words)
The last fact follows from f(X) being compact Hausdorff, and hence (since compact metrisable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrisable exactly in case they are second countable. (38 words)
The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness ( Tychonoff's theorem ) and the T 2 axiom are preserved under arbitrary products. (37 words)
Example sentences (20)
Although Hausdorff spaces are not, in general, regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular.
Although Hausdorff spaces aren't generally regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular.
However, the compact Hausdorff spaces are " absolutely closed ", in the sense that, if you embed a compact Hausdorff space K in an arbitrary Hausdorff space X, then K will always be a closed subset of X; the "surrounding space" does not matter here.
The last fact follows from f(X) being compact Hausdorff, and hence (since compact metrisable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrisable exactly in case they are second countable.
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.
Most of the time, these results hold for all preregular spaces; they were listed for regular and Hausdorff spaces separately because the idea of preregular spaces came later.
The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness ( Tychonoff's theorem ) and the T 2 axiom are preserved under arbitrary products.
Equivalent statements This theorem is equivalent to the Urysohn's lemma (which is also equivalent to the normality of the space) and is widely applicable, since all metric spaces and all compact Hausdorff spaces are normal.
However, often topological spaces must be Hausdorff spaces where limit points are unique.
Similarly locally compact Tychonoff spaces are usually just referred to as locally compact Hausdorff spaces.
There are many results for topological spaces that hold for both regular and Hausdorff spaces.
This equivalence is a consequence of the facts that compact subsets of Hausdorff spaces are closed, and closed subsets of compact spaces are compact.
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.
Another nice property of Hausdorff spaces is that compact sets are always closed.
As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces.
Embeddings into compact Hausdorff spaces may be of particular interest.
In fact, for Hausdorff spaces, paracompactness and full normality are equivalent.
One can check as an exercise that this provides an open refinement, since paracompact Hausdorff spaces are regular, and since is locally finite.
On the other hand, those results that are truly about regularity generally don't also apply to nonregular Hausdorff spaces.
Steen Seebach (1978) p.23 Paracompact Hausdorff spaces are normal.