Information AboutNormal Space |
| CATEGORIES ABOUT NORMAL SPACE | |
| topology | |
| separation axioms | |
| properties of topological spaces | |
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In Topology and related branches of Mathematics , normal spaces, '''T4 spaces''', and '''T5 spaces''' are particularly nice kinds of Topological Space s. These conditions are examples of Separation Axiom s. DEFINITIONS Suppose that ''X'' is a topological space. ''X'' is a ''normal space'' Iff , Given Any Disjoint Closed Set s ''E'' and ''F'', There Are Neighbourhood s ''U'' of ''E'' and ''V'' of ''F'' that are also disjoint. In fancier terms, this condition says that ''E'' and ''F'' can be Separated By Neighbourhoods . ''X'' is a ''T4 space'', if it's both normal and Hausdorff . ''X'' is a ''completely normal space'' or a ''hereditarily normal space'' if every Subspace of ''X'' is normal. It turns out that ''X'' is completely normal If And Only If every two Separated Set s can be separated by neighbourhoods. ''X'' is a ''T5 space'', or ''completely T4 space'', if it's both completely normal and Hausdorff, or equivalently, if every subspace of ''X'' is T4. ''X'' is a ''perfectly normal space'' if every two disjoint closed sets can be precisely separated by a function. That is, given disjoint closed sets ''E'' and ''F'', there is a Continuous Function ''f'' from ''X'' to the Real Line R such the Preimage s of {0} and {1} under ''f'' are ''E'' and ''F'' respectively. You can also use the Unit Interval {Link without Title} in this definition; the result is the same. It turns out that ''X'' is perfectly normal if and only if ''X'' is normal and every closed set is a G-delta Set . Every perfectly normal space is automatically completely normal. ''X'' is a ''perfectly T4 space'' if it is both perfectly normal and Hausdorff. Note that some mathematical literature uses different definitions for the terms "normal" and "T4", and the terms containing those words. The definitions that we have given here are the ones usually used today, and the ones used in Wikipedia. However, some authors switch the meanings of the two terms in a given pair, or use both terms synonymously for only one condition, and you should take care to find out which definitions the author is using when reading mathematical literature. (But "T5" always means the same as "completely T4", whatever that may be.) For more on this issue, see History Of The Separation Axioms . You'll also find terms like ''normal Regular space'' and ''normal Hausdorff space''; these simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T4 space. These phrases are useful, since they're less ambiguous given the historical confusion of the terms' meanings. In Wikipedia, we prefer these phrases when applicable; that is, "normal Hausdorff" instead of "T4", or "completely normal Hausdorff" instead of "T5". Fully Normal Space s and Fully T4 Space s are discussed elsewhere; they are related to Paracompactness . A Locally Normal Space is a topological space where every point has an open neighbourhood that is normal. Every normal space is locally normal, but the converse is not true. A classical example of a completely regular locally normal space that is not normal is the Niemitzki Plane . EXAMPLES OF NORMAL SPACES Most spaces encountered in Mathematical Analysis are normal Hausdorff spaces, or at least normal regular spaces:
Also, all Fully Normal Space s are normal (even if not regular). Sierpinski Space is an example of a normal space that is not regular. EXAMPLES OF NON-NORMAL SPACES An important example of a non-normal topology is given by the Zariski Topology on an Algebraic Variety or on the Spectrum Of A Ring , which is used in Algebraic Geometry . A non-normal space of some relevance to analysis is the Topological Vector Space of all Function s from the Real Line R to itself, with the Topology Of Pointwise Convergence . More generally, a theorem of A. H. Stone states that the Product of Uncountably Many non- Compact Hausdorff spaces is never normal. PROPERTIES The main significance of normal spaces lies in the fact that they admit "enough" Continuous Real -valued Function s, as expressed by the following theorems valid for any normal space ''X'': Urysohn's Lemma : If ''A'' and ''B'' are two Disjoint closed subsets of ''X'', then there exists a continuous function ''f'' from ''X'' to the real line R such that ''f''(''x'') = 0 for all ''x'' in ''A'' and ''f''(''x'') = 1 for all ''x'' in ''B''. In fact, we can take the values of ''f'' to be entirely within the Unit Interval {Link without Title} . (In fancier terms, disjoint closed sets are not only separated by neighbourhoods, but also Separated By A Function .) More generally, the Tietze Extension Theorem : If ''A'' is a closed subset of ''X'' and ''f'' is a continuous function from ''A'' to R, then there exists a continuous function ''F'': ''X'' → R which extends ''f'' in the sense that ''F''(''x'') = ''f''(''x'') for all ''x'' in ''A''. If U is a locally finite Open Cover of a normal space ''X'', then there is a Partition Of Unity precisely subordinate to U. (This shows the relationship of normal spaces to Paracompactness .) In fact, any space that satisfies any one of these conditions must be normal. A Product of normal spaces is not necessarily normal. This fact was considered surprising when it was first proved by Robert Sorgenfrey . An example of this phenomenon is the Sorgenfrey Plane . Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone-Cech compactification (which is normal Hausdorff). A more explicit example is the Tychonoff Plank . RELATIONSHIPS TO OTHER SEPARATION AXIOM S If a normal space is R0 , then it is in fact Completely Regular . Thus, anything from "normal R0" to "normal completely regular" is the same as what we normally call ''normal regular''. Taking Kolmogorov Quotient s, we see that all normal T1 Space s are Tychonoff . These are what we normally call ''normal Hausdorff'' spaces. Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpinski Space is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal. |
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