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Difference between revisions of "Relative topology"

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''of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081040/r0810401.png" /> of a [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081040/r0810402.png" />''
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''of a subset $A$ of a [[Topological space|topological space]] $(X,\tau)$''
  
The system of intersections of all possible open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081040/r0810403.png" /> (i.e. of elements of the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081040/r0810404.png" />) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081040/r0810405.png" />. The relative topology is often called the induced topology.
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The system of intersections of all possible open subsets of $(X,\tau)$ (i.e. of elements of the topology $\tau$) with $A$. The relative topology is often called the induced topology.
  
A subset of the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081040/r0810406.png" /> equipped with the relative topology is called a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081040/r0810407.png" />. A subspace of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081040/r0810408.png" />-space is itself a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081040/r0810409.png" />-space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081040/r08104010.png" /> (cf. [[Separation axiom|Separation axiom]]). A subspace of a [[Metrizable space|metrizable space]] is itself metrizable. Any Tikhonov space of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081040/r08104011.png" /> is homeomorphic to a subspace of a Hausdorff compactum of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081040/r08104012.png" /> (Tikhonov's theorem).
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A subset of the topological space $(X,\tau)$ equipped with the relative topology is called a subspace of $(X,\tau)$. A subspace of a $T_i$-space is itself a $T_i$-space, $i=0,1,2,3,31/2$ (cf. [[Separation axiom|Separation axiom]]). A subspace of a [[Metrizable space|metrizable space]] is itself metrizable. Any Tikhonov space of weight $\leq\theta$ is homeomorphic to a subspace of a Hausdorff compactum of weight $\leq\theta$ (Tikhonov's theorem).
  
  

Revision as of 19:29, 28 April 2014

of a subset $A$ of a topological space $(X,\tau)$

The system of intersections of all possible open subsets of $(X,\tau)$ (i.e. of elements of the topology $\tau$) with $A$. The relative topology is often called the induced topology.

A subset of the topological space $(X,\tau)$ equipped with the relative topology is called a subspace of $(X,\tau)$. A subspace of a $T_i$-space is itself a $T_i$-space, $i=0,1,2,3,31/2$ (cf. Separation axiom). A subspace of a metrizable space is itself metrizable. Any Tikhonov space of weight $\leq\theta$ is homeomorphic to a subspace of a Hausdorff compactum of weight $\leq\theta$ (Tikhonov's theorem).


Comments

References

[a1] J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 50ff
How to Cite This Entry:
Relative topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relative_topology&oldid=31958
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article