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Difference between revisions of "Irreducible topological space"

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A [[Topological space|topological space]] that cannot be represented as the union of two proper closed subspaces. Equivalently, an irreducible topological space can also be defined by postulating that any open subset of it is connected or that any non-empty open subset is everywhere dense. The image of an irreducible topological space under a continuous mapping is irreducible. A product of irreducible topological spaces is irreducible. The concept of an irreducible topological space is of interest only for non-separable spaces (cf. [[Separable space|Separable space]]); it is constantly used in algebraic geometry, which is concerned with the non-separable [[Zariski topology|Zariski topology]].
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An irreducible component of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052640/i0526401.png" /> is any maximal irreducible subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052640/i0526402.png" />. The irreducible components are closed and their union is the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052640/i0526403.png" />.
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''hyperconnected''
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A [[topological space]] that cannot be represented as the union of two proper closed subspaces. Equivalently, an irreducible topological space can also be defined by postulating that any open subset of it is connected or that any non-empty open subset is [[ Everywhere-dense set|everywhere dense]]. The image of an irreducible topological space under a continuous mapping is irreducible. A product of irreducible topological spaces is irreducible. The concept of an irreducible topological space is of interest only for non-separable spaces (cf. [[Separable space]]); it is constantly used in algebraic geometry, which is concerned with the non-separable [[Zariski topology]].
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An irreducible component of a topological space $X$ is any maximal irreducible subset of $X$. The irreducible components are closed and their union is the whole of $X$.
  
  
  
 
====Comments====
 
====Comments====
In the theory of coverings (cf. [[Covering (of a set)|Covering (of a set)]]) one also has a notion of irreducibility: A topological space is irreducible if every open covering of it has an irreducible open refinement, where a covering is irreducible if no proper subfamily of it is covering. Countably-compact spaces (cf. [[Countably-compact space|Countably-compact space]]) are characterized by the condition that every irreducible open covering is finite. Thus, a space is compact if and only if it is countably compact and irreducible.
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In the theory of coverings (cf. [[Covering (of a set)]]) one also has a notion of irreducibility: A topological space is irreducible if every open covering of it has an irreducible open refinement, where a covering is irreducible if no proper subfamily of it is covering. [[Countably-compact space]]s are characterized by the condition that every irreducible open covering is finite. Thus, a space is compact if and only if it is countably compact and irreducible.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Arens, J. Dugundji, "Remark on the concept of compactness" ''Portugaliae Math.'' , '''9''' (1950) pp. 141–143 {{MR|0038642}} {{ZBL|0039.18602}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Arens, J. Dugundji, "Remark on the concept of compactness" ''Portugaliae Math.'' , '''9''' (1950) pp. 141–143 {{MR|0038642}} {{ZBL|0039.18602}} </TD></TR></table>
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[[Category:General topology]]

Latest revision as of 08:04, 3 January 2016


hyperconnected

A topological space that cannot be represented as the union of two proper closed subspaces. Equivalently, an irreducible topological space can also be defined by postulating that any open subset of it is connected or that any non-empty open subset is everywhere dense. The image of an irreducible topological space under a continuous mapping is irreducible. A product of irreducible topological spaces is irreducible. The concept of an irreducible topological space is of interest only for non-separable spaces (cf. Separable space); it is constantly used in algebraic geometry, which is concerned with the non-separable Zariski topology.

An irreducible component of a topological space $X$ is any maximal irreducible subset of $X$. The irreducible components are closed and their union is the whole of $X$.


Comments

In the theory of coverings (cf. Covering (of a set)) one also has a notion of irreducibility: A topological space is irreducible if every open covering of it has an irreducible open refinement, where a covering is irreducible if no proper subfamily of it is covering. Countably-compact spaces are characterized by the condition that every irreducible open covering is finite. Thus, a space is compact if and only if it is countably compact and irreducible.

References

[a1] R. Arens, J. Dugundji, "Remark on the concept of compactness" Portugaliae Math. , 9 (1950) pp. 141–143 MR0038642 Zbl 0039.18602
How to Cite This Entry:
Irreducible topological space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_topological_space&oldid=23869
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article