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Difference between revisions of "Relative topology"

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''of a subset $A$ of a [[Topological space|topological space]] $(X,\tau)$''
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{{TEX|done}}{{MSC|54B05}}
  
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.
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''of a subset $A$ of a [[topological space]] $(X,\tau)$''
  
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).
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The system $\tau_A$ of intersections of all possible open subsets of $(X,\tau)$ (i.e. of elements of the topology $\tau$) with $A$:
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$$
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\tau_A = \{ A \cap U : U \in \tau \} \ .
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$$
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The relative topology is often called the induced topology or subspace topology. 
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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, $i=0,1,2,3,3\frac{1}{2}$ (cf. [[Separation axiom]]). A subspace of a [[metrizable space]] is itself metrizable. Any [[Tikhonov space]] of [[Weight of a topological space|weight]] $\leq\theta$ (that is, having an open [[base]] of cardinality $\leq \theta$) is homeomorphic to a subspace of a Hausdorff compactum of weight $\leq\theta$ by [[Tikhonov theorem|Tikhonov's theorem]].
  
  
  
 
====Comments====
 
====Comments====
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The subspace topology is the coarsest topology on $A$ for which the embedding map $A \hookrightarrow X$ is continuous. 
  
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Topological properties which pass to subspaces are called ''hereditary''.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , v. Nostrand  (1955)  pp. 50ff</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , v. Nostrand  (1955)  pp. 50ff</TD></TR>
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</table>

Latest revision as of 18:02, 11 July 2024

2020 Mathematics Subject Classification: Primary: 54B05 [MSN][ZBL]

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

The system $\tau_A$ of intersections of all possible open subsets of $(X,\tau)$ (i.e. of elements of the topology $\tau$) with $A$: $$ \tau_A = \{ A \cap U : U \in \tau \} \ . $$ The relative topology is often called the induced topology or subspace 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, $i=0,1,2,3,3\frac{1}{2}$ (cf. Separation axiom). A subspace of a metrizable space is itself metrizable. Any Tikhonov space of weight $\leq\theta$ (that is, having an open base of cardinality $\leq \theta$) is homeomorphic to a subspace of a Hausdorff compactum of weight $\leq\theta$ by Tikhonov's theorem.


Comments

The subspace topology is the coarsest topology on $A$ for which the embedding map $A \hookrightarrow X$ is continuous.

Topological properties which pass to subspaces are called hereditary.

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