comparison of filters (original) (raw)

Let 𝔽1 and 𝔽2 be two filters on the same set. The following terminology is commonly used to describe the relation of 𝔽1 to 𝔽2:

𝔽2 is said to be finer than 𝔽1 if 𝔽1⊆𝔽2.

𝔽2 is said to be coarser than 𝔽1 if 𝔽1⊇𝔽2.

𝔽2 is said to be strictly finer than 𝔽1 if 𝔽1⊂𝔽2.

𝔽2 is said to be strictly coarser than 𝔽1 if 𝔽1⊃𝔽2.

𝔽1 and 𝔽2 are said to be comparable if either 𝔽1⊆𝔽2 or 𝔽1⊇𝔽2.