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CLRg property

A common-limit-in-the-range condition for self-maps f and g: some sequence x_k has both f(x_k) and g(x_k) converge to the same point g(x), placing the shared limit in the range of g without requiring that range to be closed.

Version
v1 · 2026-09-28 · History
Domain-specific #
8506
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Fixed Point Theory, Metric Spaces → Mathematics

Core Idea

The CLRg property—common limit in the range of g—is an existential convergence condition on two self-maps. A pair f,g satisfies it if some sequence x_k has f(x_k) and g(x_k) converging to the same point, and that point is g(x) for some x in the space. The attained-range clause is the point of the definition.

How would you explain it like I'm…

 

No faithful explanation at this level. Two of three generators judged eli5 unreachable: a five-year-old picture of two things heading to the same spot collapses into them meeting at a common or fixed point, exactly what CLRg does not assert; the property only concerns a shared limit of a sequence that lies in g's range. A's lone two-robots version was the minority.

Common Limit Reached by g

The CLRg property is about two functions, f and g, that take points of a space to points of the same space. It asks for a sequence of inputs where the outputs of f and the outputs of g both get closer and closer to one common point. On top of that, this common point must actually be an output of g, meaning g(x) equals it for some input x. The name stands for 'common limit in the range of g'. It is a helpful step in some proofs about fixed points, but by itself it does not prove that f and g agree anywhere.

Common Limit in the Range of g

The CLRg property, short for 'common limit in the range of g', is a condition on two self-maps f and g of a space X. The pair satisfies it if there is a sequence x_k such that f(x_k) and g(x_k) both converge to the same point t, and that point is actually of the form g(x) for some x in X. The key part is the second condition, that the limit lies in the range of g. It lets fixed-point arguments go through without assuming the whole image g(X) is a closed set; only the particular common limit needs to be hit by g. CLRg by itself is not a fixed-point theorem: it does not claim f(x) = g(x) = x for any x, and further conditions are needed to reach that conclusion.

 

The CLRg property (common limit in the range of g) is an existential convergence condition on a pair of self-maps f, g of a space X: there exists a sequence (x_k) in X with lim f(x_k) = lim g(x_k) = g(x) for some x ∈ X. The attained-range clause is the substance of the definition. Fixed-point arguments often need the common limit to be realized by g; rather than assuming the whole range g(X) is closed, CLRg asks only that the relevant shared limit lie in g(X). It is therefore a weaker, targeted hypothesis rather than a global closedness assumption. CLRg by itself is not a fixed-point theorem: it does not assert that f(x) = g(x), let alone f(x) = g(x) = x; further hypotheses are needed to reach such conclusions.

Scope of Application

Use CLRg only after naming the two maps, a witness sequence, both image limits, and the point whose g-image attains the limit. Use CLRg only after naming the two maps, a witness sequence, both image limits, and the point whose g-image attains the limit.

  • Fixed-point theory. Supplies a convergence hypothesis.
  • Metric spaces. Provides sequence convergence.
  • Fuzzy metric work. Extends contractive-map arguments in the cited context.
  • Nonclosed ranges. Avoids a global closedness assumption.
  • Map compatibility studies. Compares distinct hypotheses carefully.

Clarity

Convergence of f(x_k) and g(x_k) to one point is not enough if that point is merely approached by g-values. Writing it explicitly as g(x) verifies range attainment. The closest near miss sets the boundary: A common limit in the closure of g(X) is closest: CLRg additionally requires that the limit equal an actual value g(x).

Manages Complexity

The property compresses four existential choices—maps, sequence, point, and limit—into a reusable hypothesis. It guarantees none of uniqueness, contraction, or fixedness by itself. The central weak global assumption–strong local witness tradeoff is this: No closed range is required, but the one relevant limit must be attained. A second existence–constructibility tension matters because The property asserts a sequence and point without necessarily supplying an algorithm to find them.

Abstract Reasoning

Use three linked moves: specify the carrier and convergence notion; name f and g as self-maps; exhibit a single common witness sequence. As a collapse test, the case exits when the image limits differ, convergence fails, or the shared limit is not in g's range. A fourth check is to compute both image-sequence limits.

Knowledge Transfer

Common-limit-plus-attainment reasoning transfers to operator and approximation arguments, but the CLRg name and designated range of g are fixed-point-theory specific. The nearest stopping boundary is explicit: A common limit in the closure of g(X) is closest: CLRg additionally requires that the limit equal an actual value g(x). The inclusion test remains: The pair (f,g) has CLRg when there exist x_k and x such that f(x_k) and g(x_k) both converge to the attained value g(x). The structure no longer applies when the case exits when the image limits differ, convergence fails, or the shared limit is not in g's range. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Both image sequences approach one point.

Neighborhood in Abstraction Space

CLRg property sits in a moderately populated region (51st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Abstract Algebra & Category Theory (24 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08