History[edit]

The concept of a net was first introduced by E. H. Moore and Herman L. Smith in 1922.^[1] The term "net" was coined by John L. Kelley.^[2]^[3]

The related concept of a filter was developed in 1937 by Henri Cartan.

Examples[edit]

Subspace topology[edit]

If the set $S=\{x\}\cup \left\{x_{a}:a\in A\right\}$ is endowed with the subspace topology induced on it by $X,$ then $\lim _{}x_{\bullet }\to x$ in $X$ if and only if $\lim _{}x_{\bullet }\to x$ in $S.$ In this way, the question of whether or not the net $x_{\bullet }$ converges to the given point $x$ depends solely on this topological subspace $S$ consisting of $x$ and the image of (that is, the points of) the net $x_{\bullet }.$

Characterizations of the category of topological spaces

– Family of sets representing "large" sets

Filter (set theory)

– Use of filters to describe and characterize all basic topological notions and results.

Filters in topology

– Reflexive and transitive binary relation

Preorder

– Topological space characterized by sequences

Sequential space

– Maximal proper filter

Ultrafilter (set theory)

Sundström, Manya Raman (2010). "A pedagogical history of compactness". :1006.4131v1 [math.HO].

arXiv

; Border, Kim C. (2006). Infinite dimensional analysis: A hitchhiker's guide (3rd ed.). Berlin: Springer. pp. xxii, 703. ISBN 978-3-540-32696-0. MR 2378491.

Aliprantis, Charalambos D.

Beer, Gerald (1993). Topologies on closed and closed convex sets. Mathematics and its Applications 268. Dordrecht: Kluwer Academic Publishers Group. pp. xii, 340. 0-7923-2531-1. MR 1269778.

ISBN

(23 June 1995). Modern Analysis and Topology. Graduate Texts in Mathematics. New York: Springer-Verlag Science & Business Media. ISBN 978-0-387-97986-1. OCLC 31969970. OL 1272666M.

Howes, Norman R.

(1975). General Topology. Graduate Texts in Mathematics. Vol. 27. New York: Springer Science & Business Media. ISBN 978-0-387-90125-1. OCLC 338047.

Kelley, John L.

(1991). General Topology. Springer. ISBN 3-540-90125-6.

Kelley, John L.

(1998). An Introduction to Banach Space Theory. Graduate Texts in Mathematics. Vol. 193. New York: Springer. ISBN 0-387-98431-3.

Megginson, Robert E.

(1997). Handbook of Analysis and Its Foundations. San Diego: Academic Press. ISBN 9780080532998. Retrieved 22 June 2013.

Schechter, Eric

(1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.

Schechter, Eric

Willard, Stephen (2004) [1970]. . Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.