Subnet (mathematics)
Encyclopedia
In topology
and related areas of mathematics
, a subnet is a generalization of the concept of subsequence
to the case of net
s. The definition is not completely straightforward, but is designed to allow as many theorems about subsequences to generalize to nets as possible.
If (xα) and (yβ) are nets from directed set
s A and B respectively, then (yβ) is a subnet of (xα) if there exists a monotone
cofinal function
such that
A function h : B → A is monotone if β1 ≤ β2 implies h(β1) ≤ h(β2) and cofinal if its image
is cofinal in A—that is, for every α in A there exists a β in B such that h(β) ≥ α.
While complicated, the definition does generalize some key theorems about subsequences:
A more natural definition of a subnet would be to require B to be a cofinal subset of A and that h be the identity map. This concept, known as a cofinal subnet, turns out to be inadequate. For example, the second theorem above fails for the Tychonoff plank
if we restrict ourselves to cofinal subnets.
Note that while a sequence
is a net, a sequence has subnets that are not subsequences. For example the net (1, 1, 2, 3, 4, ...) is a subnet of the net (1, 2, 3, 4, ...). The key difference is that subnets can use the same point in the net multiple times and the indexing set of the subnet can have much larger cardinality. A sequence is a subnet of a given sequence, if and only if it can be obtained from some subsequence by repeating its terms and reordering them.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and related areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a subnet is a generalization of the concept of subsequence
Subsequence
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements...
to the case of net
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...
s. The definition is not completely straightforward, but is designed to allow as many theorems about subsequences to generalize to nets as possible.
If (xα) and (yβ) are nets from directed set
Directed set
In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
s A and B respectively, then (yβ) is a subnet of (xα) if there exists a monotone
Monotonic function
In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
cofinal function
- h : B → A
such that
- yβ = xh(β).
A function h : B → A is monotone if β1 ≤ β2 implies h(β1) ≤ h(β2) and cofinal if its image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
is cofinal in A—that is, for every α in A there exists a β in B such that h(β) ≥ α.
While complicated, the definition does generalize some key theorems about subsequences:
- A net (xα) converges to x if and only if every subnet of (xα) converges to x.
- A net (xα) has a cluster point y if and only if it has a subnet (yβ) that converges to y.
- A topological space X is compactCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
if and only if every net in X has a convergent subnet (see Net (mathematics)Net (mathematics)In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...
for a proof).
A more natural definition of a subnet would be to require B to be a cofinal subset of A and that h be the identity map. This concept, known as a cofinal subnet, turns out to be inadequate. For example, the second theorem above fails for the Tychonoff plank
Tychonoff plank
In topology, the Tychonoff plank is a topological space that is a counterexample to several plausible-sounding conjectures. It is defined as the product of the two ordinal space[0,\omega_1]\times[0,\omega]...
if we restrict ourselves to cofinal subnets.
Note that while a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
is a net, a sequence has subnets that are not subsequences. For example the net (1, 1, 2, 3, 4, ...) is a subnet of the net (1, 2, 3, 4, ...). The key difference is that subnets can use the same point in the net multiple times and the indexing set of the subnet can have much larger cardinality. A sequence is a subnet of a given sequence, if and only if it can be obtained from some subsequence by repeating its terms and reordering them.