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In any topological space X, the empty set is open by definition, as is X. Since the complement of an open set is closed and the empty set and X are complements of each other, the empty set is also closed, making it a clopen set. Moreover, the empty set is compact by the fact that every finite set is compact. The closure of the empty set is empty.
Furthermore, one sometimes considers set theories in which there are no infinite sets, and then the axiom of empty set may still be required. However, any axiom of set theory or logic that implies the existence of any set will imply the existence of the empty set, if one has the axiom schema of separation. This is true, since the empty set is a ...
This perspective of niche allows for the existence of both ecological equivalents and empty niches. An ecological equivalent to an organism is an organism from a different taxonomic group exhibiting similar adaptations in a similar habitat, an example being the different succulents found in American and African deserts, cactus and euphorbia ...
A vacant niche or empty niche is an ecological niche in a particular ecosystem that is not occupied by a particular species. The issue of what exactly defines a vacant niche and whether they exist in ecosystems is controversial. The subject is intimately tied into a much broader debate on whether ecosystems can reach equilibrium, where they ...
In set theory, it may stand for "the union of x and y". Function symbols of valence 0 are called constant symbols, and are often denoted by lowercase letters at the beginning of the alphabet such as a, b and c. The symbol a may stand for Socrates. In arithmetic, it may stand for 0. In set theory, it may stand for the empty set.
Cell Biology in "The Biology Project" of University of Arizona. Centre of the Cell online; The Image & Video Library of The American Society for Cell Biology Archived 2011-06-10 at the Wayback Machine, a collection of peer-reviewed still images, video clips and digital books that illustrate the structure, function and biology of the cell.
In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set should not be confused with the empty set as defined in set theory.
In psychology, a set is a group of expectations that shape experience by making people especially sensitive to specific kinds of information. A perceptual set, also called perceptual expectancy, is a predisposition to perceive things in a certain way. [1] Perceptual sets occur in all the different senses. [2]