Definition: Comparison of Filters, Finer and Coarser Filters

If $F$ and $G$ are filters defined on the same set $X.$ Using the subset and proper subset relation, we say * $F$ is coarser than $G,$ if $F\subseteq G,$ * $F$ is strictly coarser than $G,$ if $F\subset G.$ * $F$ is finer than $G,$ if $F\supseteq G,$ * $F$ is strictly finer than $G,$ if $F\supset G,$

If one filter is finer than the other, we say that the filters are comparable, otherwise non-comparable.

Definitions: 1

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  1. Gericke, Helmuth: "Mathematik in Antike, Orient und Abendland", fourierverlag, 2003, 6th Edition
  2. Grotemeyer, K.P.: "Topologie", B.I.-Wissenschaftsverlag, 1969