Difference between revisions of "Myope topology"
From Encyclopedia of Mathematics
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A topology on the family $\mathcal{K} = \mathcal{K}_X$ of compact subsets of a topological space $X$, an instance of a "[[hit-or-miss topology]]". Let $\mathcal{F}$ denote the family of closed sets in $X$ and $\mathcal{G}$ the family of open sets. A basic open set for the myope topology is a set $U_{F,G} \subset \mathcal{K}$ of the form | A topology on the family $\mathcal{K} = \mathcal{K}_X$ of compact subsets of a topological space $X$, an instance of a "[[hit-or-miss topology]]". Let $\mathcal{F}$ denote the family of closed sets in $X$ and $\mathcal{G}$ the family of open sets. A basic open set for the myope topology is a set $U_{F,G} \subset \mathcal{K}$ of the form | ||
$$ | $$ | ||
| − | U_{F,G} = \{ A \in \mathcal{K} | + | U_{F,G} = \{ A \in \mathcal{K} : A \cap F = \emptyset\,\ A \cap G \ne \emptyset \} |
$$ | $$ | ||
where $F \in \mathcal{F}$ and $G \in \mathcal{G}$. | where $F \in \mathcal{F}$ and $G \in \mathcal{G}$. | ||
====References==== | ====References==== | ||
| − | * C. van den Berg, J. P. R. Christensen, P. Ressel, "Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions" Graduate Texts in Mathematics '''100''' | + | * C. van den Berg, J. P. R. Christensen, P. Ressel, "Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions" Graduate Texts in Mathematics '''100''' Springer (2012) {{ISBN|146121128X}} |
Latest revision as of 14:25, 12 November 2023
2020 Mathematics Subject Classification: Primary: 54B20 Secondary: 54J05 [MSN][ZBL]
A topology on the family $\mathcal{K} = \mathcal{K}_X$ of compact subsets of a topological space $X$, an instance of a "hit-or-miss topology". Let $\mathcal{F}$ denote the family of closed sets in $X$ and $\mathcal{G}$ the family of open sets. A basic open set for the myope topology is a set $U_{F,G} \subset \mathcal{K}$ of the form $$ U_{F,G} = \{ A \in \mathcal{K} : A \cap F = \emptyset\,\ A \cap G \ne \emptyset \} $$ where $F \in \mathcal{F}$ and $G \in \mathcal{G}$.
References
- C. van den Berg, J. P. R. Christensen, P. Ressel, "Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions" Graduate Texts in Mathematics 100 Springer (2012) ISBN 146121128X
How to Cite This Entry:
Myope topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Myope_topology&oldid=51365
Myope topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Myope_topology&oldid=51365