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SCHMIDT'S GAME ON HAUSDORFF METRIC AND FUNCTION SPACES: GENERIC DIMENSION OF SETS AND IMAGES
Author(s) -
Farkas Ábel,
Fraser Jonathan M.,
Nesharim Erez,
Simmons David
Publication year - 2021
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/mtk.12068
Subject(s) - mathematics , dimension (graph theory) , hausdorff space , metric space , mathematical economics , combinatorics , pure mathematics
We consider Schmidt's game on the space of compact subsets of a given metric space equipped with the Hausdorff metric, and the space of continuous functions equipped with the supremum norm. We are interested in determining the generic behavior of objects in a metric space, mostly in the context of fractal dimensions, and the notion of “generic” we adopt is that of being winning for Schmidt's game. We find properties whose corresponding sets are winning for Schmidt's game that are starkly different from previously established, and well‐known, properties which are generic in other contexts, such as being residual or of full measure.