Difference between revisions of "Relative topology"
From Encyclopedia of Mathematics
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| − | ''of a subset $A$ of a [[ | + | ''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 | + | 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, $i=0,1,2,3,31/2$ (cf. [[ | + | 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). |
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====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> | <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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| + | [[Category:General topology]] | ||
Revision as of 12:34, 9 November 2014
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, $i=0,1,2,3,31/2$ (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=31959
Relative topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relative_topology&oldid=31959
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article