View source for Relative topology

{{TEX|done}}{{MSC|54B05}}

''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====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , v. Nostrand  (1955)  pp. 50ff</TD></TR>
</table>