Subspace topology

In mathematical branch of topology is meant by the subspace topology (also induced topology , relative topology , track topology or subspace topology ) the natural structure, which is a subset of a topological space "inherits". The subspace topology is a special initial topology .

Formal definition

Let it be the basic set of a topological space and a subset. Then the subspace topology is based on the topology ${\displaystyle X}$ ${\displaystyle \left(X,{\mathcal {T}}\right)}$ ${\displaystyle Y\subseteq X}$ ${\displaystyle Y}$

${\displaystyle {\mathcal {T}}_{Y}=\{O\cap Y\mid O\in {\mathcal {T}}\}.}$

The open subsets of are therefore exactly the intersections of the open subsets of with . ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

properties

• The subspace topology on a subset of a topological space is the weakest topology for which the inclusion mapping ${\displaystyle Y\subseteq X}$ ${\displaystyle X}$

${\displaystyle Y\to X,\quad y\mapsto y}$

is steady .

• If an open subset of a topological space is , then a subset is open in the subspace topology of if and only if is open as a subset of . ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle U\subseteq Y}$ ${\displaystyle Y}$ ${\displaystyle U}$ ${\displaystyle X}$
• If a closed subset of a topological space is , then a subset is closed in the subspace topology of if and only if it is closed as a subset of . ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Z\subseteq Y}$ ${\displaystyle Y}$ ${\displaystyle Z}$ ${\displaystyle X}$
• A continuous mapping of topological spaces is a monomorphism in the sense of category theory if and only if it is a homeomorphism as mapping to the set-theoretical image provided with the subspace topology. In particular, monomorphisms are injective .

Examples

• Imagine a sheet of paper without a border as a two-dimensional object. Im not an open set. However, if you look at the topology with regard to the level in which the leaf is located, then there is an open set. ${\displaystyle \mathbb {R} ^{3}}$
• The subspace topology on is the discrete topology ; H. all subsets of are open as subsets of topological space . For example, the set is an open subset of because it is the intersection of the open subset of with . ${\displaystyle \mathbb {Z} \subset \mathbb {R} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \{0\}}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \left(-{\tfrac {1}{2}},{\tfrac {1}{2}}\right)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Z} }$

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .