Difference between revisions of "Relative topology"
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\tau_A = \{ A \cap U : U \in \tau \} \ . | \tau_A = \{ A \cap U : U \in \tau \} \ . | ||
$$ | $$ | ||
| − | The relative topology is often called the induced topology or subspace topology | + | 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). | 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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====Comments==== | ====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==== | ====References==== | ||
Revision as of 19:44, 3 January 2016
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, $i=0,1,2,3,31/2$ (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
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 |
Relative topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relative_topology&oldid=35547