Editing Net (mathematics) (section)

=== Neighborhood systems ===
{{Main|Neighborhood system}}

Intuitively, convergence of a net <math>\left(x_a\right)_{a \in A}</math> means that the values <math>x_a</math> come and stay as close as we want to <math>x</math> for large enough <math>a.</math> Given a point <math>x</math> in a topological space, let <math>N_x</math> denote the set of all [[Neighbourhood (topology)|neighbourhood]]s containing <math>x.</math> Then <math>N_x</math> is a directed set, where the direction is given by reverse inclusion, so that <math>S \geq T</math> [[if and only if]] <math>S</math> is contained in <math>T.</math> For <math>S \in N_x,</math> let <math>x_S</math> be a point in <math>S.</math> Then <math>\left(x_S\right)</math> is a net. As <math>S</math> increases with respect to <math>\,\geq,</math> the points <math>x_S</math> in the net are constrained to lie in decreasing neighbourhoods of <math>x,</math>. Therefore, in this [[neighborhood system]] of a point <math>x</math>, <math>x_S</math> does indeed converge to <math>x</math> according to the definition of net convergence.

Given a [[subbase]] <math>\mathcal{B}</math> for the topology on <math>X</math> (where note that every [[Base (topology)|base]] for a topology is also a subbase) and given a point <math>x \in X,</math> a net <math>x_\bull</math> in <math>X</math> converges to <math>x</math> if and only if it is eventually in every neighborhood <math>U \in \mathcal{B}</math> of <math>x.</math> This characterization extends to [[Neighbourhood system|neighborhood subbases]] (and so also [[Neighbourhood system|neighborhood bases]]) of the given point <math>x.</math>