User:Kompik/Math/Filter

Various definition of filter/ideal

In mathematical literature, various definitions of a filter/ideal can be found.

Filter defined with the condition $\emptyset \notin {\mathcal {F}}$: [B, p.57, Definition 1] [HH, p.74, Definition 9.1], [HJ, p.201, Definition 11.1], [J, Definition 7.1], [L, Definition 4.1]

Filter defined without the condition $\emptyset \notin {\mathcal {F}}$: [JW2, p.1], [C, p.60, Exercise 3]

Ideal defined with the condition $X\notin {\mathcal {I}}$: [HH, p.145, Definition 12.2], [HJ, p.202, Definition 11.1], [J, Definition 7.1], [L, Definition 4.1]

Ideal defined without the condition $X\notin {\mathcal {I}}$: [C, p.14], [JW2, p.2]

References

[B] N. Bourbaki. Elements of Mathematics. General Topology. Chapters I-IV. Springer-Verlag, Berlin, 1989.

[C] Krzysztof Ciesielski. Set Theory for the Working Mathematician. Cambridge University Press, Cambridge, 1997. London Mathematical Society Student Texts 39.

[HH] A. Hajnal and P. Hamburger. Set theory. Cambridge University Press, Cambridge, 1999.

[HJ] K. Hrbacek and T. Jech. Introduction to set theory. Marcel Dekker, New York, 1999.

[J] T. Jech. Set theory. Springer, Berlin, 2002.

[JW2] Winfried Just and Martin Weese. Discovering modern set theory II: Set-theoretic tools for every mathematician. American Mathematical Society (AMS), Providence, RI, 1997. Graduate Studies in Mathematics 18.

[L] Azriel Levy. Basic set theory. Courier Dover Publications, 2002.