Completed picture

Completed images are considered in the mathematical sub-area of topology . These are mappings between two topological spaces that map closed sets to closed sets.

definition

Let be a mapping between the topological spaces and . is said to be closed if the image set is also closed for every closed set . ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f}$ ${\displaystyle A\subset X}$ ${\displaystyle f(A)\subset Y}$

Examples

• Every continuous mapping from a bounded , closed interval into the real numbers is closed. This does not apply to unlimited intervals, for example the continuous arctangent function is not closed because it is closed, but the image set is not closed. ${\displaystyle f\colon [a,b]\rightarrow \mathbb {R} }$ ${\displaystyle \mathrm {arctan} \colon \mathbb {R} \rightarrow \mathbb {R} }$ ${\displaystyle A=[0,\infty )\subset \mathbb {R} }$ ${\displaystyle \textstyle \mathrm {arctan} (A)=[0,{\frac {\pi }{2}})\subset \mathbb {R} }$
• In general, every continuous mapping from a compact space to a Hausdorff space is closed. This is because if it is closed, then as a closed subset of a compact space is compact and therefore the image is also compact. As a compact subset of a Hausdorff area is complete. ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle A\subset X}$ ${\displaystyle A}$ ${\displaystyle f(A)}$ ${\displaystyle f(A)}$
• Homeomorphisms are complete. More precisely, a bijective mapping between topological spaces is a homeomorphism if and only if is continuous and closed. ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle f}$
• Actual illustrations are complete. A continuous mapping is more precise exactly when it is closed and compact for each . ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle f^{-1}(\{y\})\subset X}$ ${\displaystyle y\in f(X)}$
• Open images do not have to be closed. The mapping is open, the image set of the closed set is the non-closed set . Conversely, closed maps do not have to be open, as the example of a constant map shows. ${\displaystyle f\colon \mathbb {R} ^{2}\to \mathbb {R} ,\,(s,t)\mapsto s}$ ${\displaystyle \{(s,t)\colon s\geq 0,st\geq 1\}}$ ${\displaystyle (0,\infty )}$

properties

• Compositions of completed images are completed again.
• Be a closed figure and be open. Then it's open. ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle B\subset Y}$ ${\displaystyle f^{-1}(B)\subset X}$ ${\displaystyle B}$
• A mapping between topological spaces is complete if and only if for all subsets . ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle f({\overline {A}})\supset {\overline {f(A)}}}$ ${\displaystyle A\subset X}$

Demarcation

In functional analysis one considers so-called closed operators between topological vector spaces and , these are linear operators whose graph is a closed set in the product space . This must not be confused with the concept of closed mapping between topological spaces considered above. For example, the inclusion mapping of the sequential spaces with their usual norm topologies as a continuous, linear operator is definitely closed, but it is not a closed mapping between the associated topological spaces, because it is closed, but the picture is not closed. ${\displaystyle T\colon X\rightarrow Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X\times Y}$ ${\displaystyle \iota \colon \ell ^{1}\rightarrow \ell ^{\infty }}$ ${\displaystyle A=\ell ^{1}\subset \ell ^{1}}$ ${\displaystyle \iota (A)\subset \ell ^{\infty }}$

Individual evidence

1. ^ Wolfgang Franz : Topologie I. Walter de Gruyter, 1973, ISBN 3-11-004117-0 , p. 37.
2. ^ Boto von Querenburg : Set theoretical topology. 3. Edition. Springer, Berlin 2001, ISBN 3-540-67790-9 , definition 2.26.
3. ^ HJ Kowalsky : Topological Spaces. Springer-Verlag, 1961, ISBN 978-3-0348-6907-2 , definition 17b.
4. ^ Wolfgang Franz : Topologie I. Walter de Gruyter, 1973, ISBN 3-11-004117-0 , sentence 5.7.
5. ^ Boto von Querenburg : Set theoretical topology. 3. Edition. Springer, Berlin 2001, ISBN 3-540-67790-9 , sentence 2.28.
6. ↑ Erich Ossa : Topology. Vieweg + Teubner publishing house, ISBN 3-8348-0874-1 , sentence 2.4.20.
7. ↑ Dirk Werner : Functional Analysis. 6th edition. Springer-Verlag, 2007, ISBN 978-3-540-72533-6 , behind definition IV.3.1.
8. ^ HJ Kowalsky: Topological Spaces. Springer-Verlag, 1961, ISBN 978-3-0348-6907-2 , sentence 17.9.
9. ^ HJ Kowalsky: Topological Spaces. Springer-Verlag, 1961, ISBN 978-3-0348-6907-2 , sentence 17.8.