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A set with an upper (respectively, lower) bound is said to be bounded from above or majorized [1] (respectively bounded from below or minorized) by that bound. The terms bounded above ( bounded below ) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.
The metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For subsets of R n the two are equivalent. A metric space is compact if and only if it is complete and totally bounded. A subset of Euclidean space R n is compact if and only if it is closed and
In particular, every subset Y of X is bounded above by X and below by the empty set ∅ because ∅ ⊆ Y ⊆ X. Hence, it is possible (and sometimes useful) to consider superior and inferior limits of sequences in ℘(X) (i.e., sequences of subsets of X). There are two common ways to define the limit of sequences of sets. In both cases:
A real-valued function is bounded if and only if it is bounded from above and below. [ 1 ] [ additional citation(s) needed ] An important special case is a bounded sequence , where X {\displaystyle X} is taken to be the set N {\displaystyle \mathbb {N} } of natural numbers .
Taller species will have part of their shoot system in the underlying layers. In addition to the above-ground stratification there is also a “root layer”. In the broadest sense, the layering of diaspores in the soil may be counted as part of the vertical structure. The plants of a layer, especially with regard to their way of life and ...
Hypogeal germination (from Ancient Greek ὑπόγειος [hupógeios] 'below ground', from ὑπό [hupó] 'below' and γῆ [gê] 'earth, ground') is a botanical term indicating that the germination of a plant takes place below the ground. An example of a plant with hypogeal germination is the pea (Pisum sativum). The opposite of hypogeal is ...
The set S obviously contains a, and is bounded by b by construction. By the least-upper-bound property, S has a least upper bound c ∈ [a, b]. Hence, c is itself an element of some open set U α, and it follows for c < b that [a, c + δ] can be covered by finitely many U α for some sufficiently small δ > 0.
Epigeal vs. hypogeal germination. Epigeal germination (Ancient Greek ἐπίγαιος [epígaios] 'above ground', from ἐπί [epí] 'on' and γῆ [gê] 'earth, ground') is a botanical term indicating that the germination of a plant takes place above the ground. An example of a plant with epigeal germination is the common bean (Phaseolus ...