Countable setIn mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.
Axiom of countable choiceThe axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function A with domain N (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n ∈ N, there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N. The axiom of countable choice (ACω) is strictly weaker than the axiom of dependent choice (DC), which in turn is weaker than the axiom of choice (AC).
Countably compact spaceIn mathematics a topological space is called countably compact if every countable open cover has a finite subcover. A topological space X is called countably compact if it satisfies any of the following equivalent conditions: (1) Every countable open cover of X has a finite subcover. (2) Every infinite set A in X has an ω-accumulation point in X. (3) Every sequence in X has an accumulation point in X. (4) Every countable family of closed subsets of X with an empty intersection has a finite subfamily with an empty intersection.
Extinction eventAn extinction event (also known as a mass extinction or biotic crisis) is a widespread and rapid decrease in the biodiversity on Earth. Such an event is identified by a sharp change in the diversity and abundance of multicellular organisms. It occurs when the rate of extinction increases with respect to the background extinction rate and the rate of speciation. Estimates of the number of major mass extinctions in the last 540 million years range from as few as five to more than twenty.
Holocene extinctionThe Holocene extinction, or Anthropocene extinction, is the ongoing extinction event during the Holocene epoch. The extinctions span numerous families of plants and animals, including mammals, birds, reptiles, amphibians, fish, invertebrates, and affecting not just terrestrial species but also large sectors of marine life. With widespread degradation of biodiversity hotspots, such as coral reefs and rainforests, as well as other areas, the vast majority of these extinctions are thought to be undocumented, as the species are undiscovered at the time of their extinction, which goes unrecorded.
Infinite setIn set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers.
ExtinctionExtinction is the termination of a taxon by the death of its last member. A taxon may become functionally extinct before the death of its last member if it loses the capacity to reproduce and recover. Because a species' potential range may be very large, determining this moment is difficult, and is usually done retrospectively. This difficulty leads to phenomena such as Lazarus taxa, where a species presumed extinct abruptly "reappears" (typically in the fossil record) after a period of apparent absence.
Dedekind-infinite setIn mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. Explicitly, this means that there exists a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers.
First-countable spaceIn topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point in there exists a sequence of neighbourhoods of such that for any neighbourhood of there exists an integer with contained in Since every neighborhood of any point contains an open neighborhood of that point, the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods.
Second-countable spaceIn topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space is second-countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.
Ordinal numberIn set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element").
Permian–Triassic extinction eventThe Permian–Triassic (P–T, P–Tr) extinction event (PTME), also known as the Late Permian extinction event, the Latest Permian extinction event, the End-Permian extinction event, and colloquially as the Great Dying, forms the boundary between the Permian and Triassic geologic periods, and with them the Paleozoic and Mesozoic eras respectively, approximately 251.9 million years ago. As the largest of the "Big Five" mass extinctions of the Phanerozoic, it is the Earth's most severe known extinction event, with the extinction of 57% of biological families, 83% of genera, 81% of marine species and 70% of terrestrial vertebrate species.
Axiom of countabilityIn mathematics, an axiom of countability is a property of certain mathematical objects that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not provably exist.
Extinction risk from climate changeThere are several plausible pathways that could lead to an increased extinction risk from climate change. This is because every plant and animal species has evolved to exist within a certain ecological niche, and as climate change represents the long-term alteration of temperature and average weather patterns, it can push climatic conditions outside of the species' niche, which will ultimately render it extinct. Normally, species faced with changing conditions can either adapt in place through microevolution or move to another habitat with suitable conditions.
Quaternary extinction eventThe latter half of the Late Pleistocene to the beginning of the Holocene (~50,000-10,000 years Before Present) saw extinctions of numerous predominantly megafaunal species, which resulted in a collapse in faunal density and diversity across the globe. The extinctions during the Late Pleistocene are differentiated from previous extinctions by the widespread absence of ecological succession to replace these extinct megafaunal species, and the regime shift of previously established faunal relationships and habitats as a consequence.
Iterative methodIn computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method.
Capitanian mass extinction eventThe Capitanian mass extinction event, also known as the end-Guadalupian extinction event, the Guadalupian-Lopingian boundary mass extinction, the pre-Lopingian crisis, or the Middle Permian extinction was an extinction event that predated the end-Permian extinction event. The mass extinction occurred during a period of decreased species richness and increased extinction rates near the end of the Middle Permian, also known as the Guadalupian epoch.
Late Devonian extinctionThe Late Devonian extinction consisted of several extinction events in the Late Devonian Epoch, which collectively represent one of the five largest mass extinction events in the history of life on Earth. The term primarily refers to a major extinction, the Kellwasser event, also known as the Frasnian-Famennian extinction, which occurred around 372 million years ago, at the boundary between the Frasnian stage and the Famennian stage, the last stage in the Devonian Period. Overall, 19% of all families and 50% of all genera became extinct.
Human extinctionHuman extinction is the hypothetical end of the human species due to either natural causes such as population decline from sub-replacement fertility, an asteroid impact, large-scale volcanism, or via anthropogenic destruction (self-extinction). For the latter, some of the many possible contributors include climate change, global nuclear annihilation, biological warfare, and ecological collapse. Other scenarios center on emerging technologies, such as advanced artificial intelligence, biotechnology, or self-replicating nanobots.
Biodiversity lossBiodiversity loss includes the worldwide extinction of different species, as well as the local reduction or loss of species in a certain habitat, resulting in a loss of biological diversity. The latter phenomenon can be temporary or permanent, depending on whether the environmental degradation that leads to the loss is reversible through ecological restoration/ecological resilience or effectively permanent (e.g. through land loss).
