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| In [[topology]] and related areas of [[mathematics]] a '''topological property''' or '''topological invariant''' is a property of a [[topological space]] which is [[invariant (mathematics)|invariant]] under [[homeomorphism]]s. That is, a property of spaces is a topological property if whenever a space ''X'' possesses that property every space homeomorphic to ''X'' possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.
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| A common problem in topology is to decide whether two topological spaces are [[homeomorphic]] or not. To prove that two spaces are ''not'' homeomorphic, it is sufficient to find a topological property which is not shared by them.
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| ==Common topological properties==
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| === [[Cardinal function]]s ===
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| * The [[cardinality]] |X| of the space X.
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| * The cardinality τ(X) of the topology of the space X.
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| * ''Weight'' w(X), the least cardinality of a [[basis (topology)|basis of the topology]] of the space X.
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| * ''Density'' d(X), the least cardinality of a subset of X whose closure is X.
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| === Separation ===
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| For a detailed treatment, see [[separation axiom]]. Some of these terms are defined differently in older mathematical literature; see [[history of the separation axioms]].
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| * '''T<sub>0</sub>''' or '''Kolmogorov'''. A space is [[Kolmogorov space|Kolmogorov]] if for every pair of distinct points ''x'' and ''y'' in the space, there is at least either an open set containing ''x'' but not ''y'', or an open set containing ''y'' but not ''x''.
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| * '''T<sub>1</sub>''' or '''Fréchet'''. A space is [[T1 space| Fréchet]] if for every pair of distinct points ''x'' and ''y'' in the space, there is an open set containing ''x'' but not ''y''. (Compare with T<sub>0</sub>; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T<sub>1</sub> if all its singletons are closed. T<sub>1</sub> spaces are always T<sub>0</sub>.
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| * '''Sober'''. A space is [[sober space|sober]] if every irreducible closed set ''C'' has a unique generic point ''p''. In other words, if ''C'' is not the (possibly nondisjoint) union of two smaller closed subsets, then there is a ''p'' such that the closure of {''p''} equals ''C'', and ''p'' is the only point with this property.
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| * '''T<sub>2</sub>''' or '''Hausdorff'''. A space is [[Hausdorff space|Hausdorff]] if every two distinct points have disjoint neighbourhoods. T<sub>2</sub> spaces are always T<sub>1</sub>.
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| * '''T<sub>2½</sub>''' or '''Urysohn'''. A space is [[Urysohn space| Urysohn ]] if every two distinct points have disjoint ''closed'' neighbourhoods. T<sub>2½</sub> spaces are always T<sub>2</sub>.
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| * '''Regular'''. A space is [[regular space|regular]] if whenever ''C'' is a closed set and ''p'' is a point not in ''C'', then ''C'' and ''p'' have disjoint neighbourhoods.
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| * '''T<sub>3</sub>''' or '''Regular Hausdorff'''. A space is [[regular Hausdorff space|regular Hausdorff]] if it is a regular T<sub>0</sub> space. (A regular space is Hausdorff if and only if it is T<sub>0</sub>, so the terminology is [[consistent]].)
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| * '''Completely regular'''. A space is [[Tychonoff space|completely regular]] if whenever ''C'' is a closed set and ''p'' is a point not in ''C'', then ''C'' and {''p''} are [[separated by a function]].
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| * '''T<sub>3½</sub>''', '''Tychonoff''', '''Completely regular Hausdorff''' or '''Completely T<sub>3</sub>'''. A [[Tychonoff space]] is a completely regular T<sub>0</sub> space. (A completely regular space is Hausdorff if and only if it is T<sub>0</sub>, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff.
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| * '''Normal'''. A space is [[normal space|normal]] if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit [[partition of unity|partitions of unity]].
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| * '''T<sub>4</sub>''' or '''Normal Hausdorff'''. A normal space is Hausdorff if and only if it is T<sub>1</sub>. Normal Hausdorff spaces are always Tychonoff.
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| * '''Completely normal'''. A space is [[completely normal]] if any two separated sets have disjoint neighbourhoods.
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| * '''T<sub>5</sub>''' or '''Completely normal Hausdorff'''. A completely normal space is Hausdorff if and only if it is T<sub>1</sub>. Completely normal Hausdorff spaces are always normal Hausdorff.
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| * '''Perfectly normal'''. A space is [[perfectly normal space|perfectly normal]] if any two disjoint closed sets are [[precisely separated by a function]]. A perfectly normal space must also be completely normal.
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| * '''Perfectly normal Hausdorff''', or '''perfectly T<sub>4</sub>'''. A space is [[perfectly normal Hausdorff space|perfectly normal Hausdorff]], if it is both perfectly normal and T<sub>1</sub>. A perfectly normal Hausdorff space must also be completely normal Hausdorff.
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| * '''Discrete space'''. A space is [[discrete space|discrete]] if all of its points are completely isolated, i.e. if any subset is open.
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| === Countability conditions ===
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| * '''Separable'''. A space is [[separable (topology)|separable]] if it has a [[countable]] dense subset.
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| * '''Lindelöf'''. A space is [[Lindelöf space|Lindelöf]] if every open cover has a [[countable]] subcover.
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| * '''First-countable'''. A space is [[first-countable space|first-countable]] if every point has a [[countable]] local base.
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| * '''Second-countable'''. A space is [[second-countable space|second-countable]] if it has a [[countable]] base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.
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| === Connectedness ===
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| * '''Connected'''. A space is [[Connected space|connected]] if it is not the union of a pair of disjoint non-empty open sets. Equivalently, a space is connected if the only [[clopen set]]s are the empty set and itself.
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| * '''Locally connected'''. A space is [[locally connected]] if every point has a local base consisting of connected sets.
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| * '''Totally disconnected'''. A space is [[totally disconnected]] if it has no connected subset with more than one point.
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| * '''Path-connected'''. A space ''X'' is [[path-connected]] if for every two points ''x'', ''y'' in ''X'', there is a path ''p'' from ''x'' to ''y'', i.e., a continuous map ''p'': [0,1] → ''X'' with ''p''(0) = ''x'' and ''p''(1) = ''y''. Path-connected spaces are always connected.
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| * '''Locally path-connected'''. A space is [[locally path-connected]] if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected.
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| * '''Simply connected'''. A space ''X'' is [[simply connected]] if it is path-connected and every continuous map ''f'': S<sup>1</sup> → ''X'' is [[homotopic]] to a constant map.
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| *'''Locally simply connected'''. A space ''X'' is [[locally simply connected space|locally simply connected]] if every point ''x'' in ''X'' has a local base of neighborhoods ''U'' that is simply connected.
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| *'''Semi-locally simply connected'''. A space ''X'' is [[semi-locally simply connected]] if every point has a local base of neighborhoods ''U'' such that ''every'' loop in ''U'' is contractible in ''X''. Semi-local simple connectivity, a strictly weaker condition than local simple connectivity, is a necessary condition for the existence of a [[universal cover]].
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| * '''Contractible'''. A space ''X'' is contractible if the [[identity function|identity map]] on ''X'' is homotopic to a constant map. Contractible spaces are always simply connected.
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| * '''Hyper-connected'''. A space is [[hyper-connected]] if no two non-empty open sets are disjoint. Every hyper-connected space is connected.
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| * '''Ultra-connected'''. A space is [[ultra-connected]] if no two non-empty closed sets are disjoint. Every ultra-connected space is path-connected.
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| * '''Indiscrete''' or '''trivial'''. A space is [[indiscrete space|indiscrete]] if the only open sets are the empty set and itself. Such a space is said to have the [[trivial topology]].
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| === Compactness ===
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| * '''Compact'''. A space is [[Compact space|compact]] if every [[open cover]] has a finite subcover. Some authors call these spaces '''quasicompact''' and reserve compact for [[Hausdorff space|Hausdorff]] spaces where every open cover has finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal.
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| * '''Sequentially compact'''. A space is [[sequentially compact]] if every sequence has a convergent subsequence.
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| * '''Countably compact'''. A space is [[countably compact]] if every countable open cover has a finite subcover.
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| * '''Pseudocompact'''. A space is [[pseudocompact]] if every continuous real-valued function on the space is bounded.
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| * '''σ-compact'''. A space is [[σ-compact space|σ-compact]] if it is the union of countably many compact subsets.
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| * '''Paracompact'''. A space is [[paracompact]] if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal.
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| * '''Locally compact'''. A space is [[locally compact]] if every point has a local base consisting of compact neighbourhoods. Slightly different definitions are also used. Locally compact Hausdorff spaces are always Tychonoff.
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| * '''Ultraconnected compact'''. In an ultra-connected compact space ''X'' every open cover must contain ''X'' itself. Non-empty ultra-connected compact spaces have a largest proper open subset called a '''monolith'''.
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| === Metrizability ===
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| * '''Metrizable'''. A space is [[Metrization theorems|metrizable]] if it is homeomorphic to a [[metric space]]. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable.
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| * '''Polish'''. A space is called Polish if it is metrizable with a separable and complete metric.
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| * '''Locally metrizable'''. A space is locally metrizable if every point has a metrizable neighbourhood.
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| === Miscellaneous ===
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| * '''Baire space'''. A space ''X'' is a [[Baire space]] if it is not [[Meagre set|meagre]] in itself. Equivalently, ''X'' is a Baire space if the intersection of countably many dense open sets is dense.
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| * '''Homogeneous'''. A space ''X'' is homogeneous if for every ''x'' and ''y'' in ''X'' there is a homeomorphism ''f'' : ''X'' → ''X'' such that ''f''(''x'') = ''y''. Intuitively speaking, this means that the space looks the same at every point. All [[topological group]]s are homogeneous.
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| * '''Finitely generated''' or '''Alexandrov'''. A space ''X'' is [[Alexandrov topology|Alexandrov]] if arbitrary intersections of open sets in ''X'' are open, or equivalently if arbitrary unions of closed sets are closed. These are precisely the [[Finitely generated object|finitely generated]] members of the [[category of topological spaces]] and continuous maps.
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| * '''Zero-dimensional'''. A space is [[zero-dimensional]] if it has a base of clopen sets. These are precisely the spaces with a small [[inductive dimension]] of ''0''.
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| * '''Almost discrete'''. A space is [[almost discrete]] if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.
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| * '''Boolean'''. A space is [[Boolean space|Boolean]] if it is zero-dimensional, compact and Hausdorff (equivalently, totally disconnected, compact and Hausdorff). These are precisely the spaces that are homeomorphic to the [[Stone space]]s of [[Boolean algebra (structure)|Boolean algebra]]s.
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| * '''[[Reidemeister torsion]]'''
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| * '''<math>\kappa</math>-resolvable'''. A space is said to be κ-resolvable<ref>{{cite journal|last=Juhász|first=István|coauthors=Soukup, Lajos; Szentmiklóssy, Zoltán|title=Resolvability and monotone normality|journal=Israel Journal of Mathematics|year=2008|volume=166|issue=1|pages=1–16|doi=10.1007/s11856-008-1017-y|url=http://link.springer.com/content/pdf/10.1007%2Fs11856-008-1017-y.pdf|accessdate=4 December 2012|publisher=The Hebrew University Magnes Press|issn=0021-2172}}</ref> (respectively: almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (respectively: almost disjoint over the ideal of nowhere dense subsets). If the space is not <math>\kappa</math>-resolvable then it is called <math>\kappa</math>-irresolvable.
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| * '''Maximally resolvable'''. Space <math>X</math> is maximally resolvable if it is <math>\Delta(X)</math>-resolvable, where <math>\Delta(X) =
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| \min\{|G| : G\neq\emptyset, G\mbox{ is open}\}</math>. Number <math>\Delta(X)</math> is called dispersion character of <math>X</math>.
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| * '''Strongly discrete'''. Set <math>D</math> is strongly discrete subset of the space <math>X</math> if the points in <math>D</math> may be separated by pairwise disjoint neighborhoods. Space <math>X</math> is said to be strongly discrete if every non-isolated point of <math>X</math> is the [[Limit point|accumulation point]] of some strongly discrete set.
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| ==See also==
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| *[[Euler characteristic]]
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| *[[Winding number]]
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| *[[Characteristic class]]
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| *[[Characteristic numbers]]
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| *[[Chern class]]
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| *[[Knot invariant]]
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| *[[Linking number]]
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| *[[Fixed point property]]
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| *[[Topological quantum number]]
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| *[[Homotopy group]] and [[Cohomotopy group]]
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| *[[Homology (mathematics)|Homology]] and [[cohomology]]
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| *[[Quantum invariant]]
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| ==References==
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| <references/>
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| ==Bibliography==
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| * {{cite book|last=Willard|first=Stephen|title=General topology|year=1970|publisher=Addison-Wesley Pub. Co|location=Reading, Mass.|isbn=9780486434797|pages=369|url=http://books.google.com.mx/books?id=-o8xJQ7Ag2cC}}
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| [[Category:Properties of topological spaces| ]]
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| [[Category:Homeomorphisms]]
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| [[ru:Топологический инвариант]]
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Research has shown that since the amygdala gives more weight to NEGATIVE, painful experiences than positive ones, that in order for a marriage or relationship to last, you must create 4 positive experiences for each negative one. Sesame oil is great, but sunflower, coconut, or corn oil will work, too. In other words if you are looking for a quick fix (aren't we all), you may be in for a disappointment unless your next move is just right for you. Keep an eye out for my article, Panic Attacks and Anxiety: Adios. To some, these new regulations may seem intrusive, but they were instilled with the best of intentions.
Although this disorder reacts to everyone differently, it can be very frustrating to live your life with it. You will have occasions when you won't be able to focus on what exactly the specific reason of your German Shepherd anxiety is. The typical symptoms of panic disorder include sudden attacks of fear and nervousness, as well as physical symptoms such as sweating and a racing heart. Your nervous feelings will dissipate if you focus your attention away from your anxieties and concentrate on your message and your audience, not yourself. Each of these offer different concepts and processes so it's up to the person which one could be suitable to his or her needs.
This is the most practical way of eliminating any kind of fear and this works well for anxiety attacks. Dogs that are well exercised spend more of their idle time resting rather than panicking. Are your worries logically justified or are you blowing things out of proportion. There are chemical messengers called neuro transmitters which transmit signals from one brain cell to another cell. But, yes, hypertension is on the rise in our society, but this is largely due to diet issues.
This is the largest step you will have to take when dealing with your anxiety. Today we use techniques that are simple, fast, relaxing and effective. As you can see anxiety problems are not uncommon in children and when identified and treated, they can be short-lived. Diseases - They follow a logical psychological progression based on our life experiences and we can map exactly what happened and why, the effect it had on us and how this fits in with our problem. I decided to do a little research about the herb and found out that it has mild tranquilizing properties.
This high motivational need for achievement is where the quest for self-actualization is realized which embodies what is considered a continual yearning and eagerness to survive. A big issue with many being affected by the common anxiety dysfunction is that with all of their thinking and being troubled, they can cause an all out attack. Use only a select couple of items of essential furnishings, and retain a lot of open space. Being stressed out, afraid and filled with anxiety are mental emotions that are common place in our daily lives. Ask yourself if there are emotional or other barriers to your progress.
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