In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states:
There is no set whose cardinality is strictly between that of the integers and the real numbers.
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.
In set theory, a Woodin cardinal is a cardinal number λ such that for all functions
In topology and related branches of mathematics, a totally disconnected space is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets. In every topological space the empty set and the one-point sets are connected; in a totally disconnected space these are the only connected subsets.
In topology, an open map is a function between two topological spaces which maps open sets to open sets. That is, a function is open if for any open set U in X, the image is open in Y. Likewise, a closed map is a function which maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.
In descriptive set theory, a subset of a Polish space is an analytic set if it is a continuous image of a Polish space. These sets were first defined by Luzin (1917) and his student Souslin (1917).
In mathematics, the axiom of determinacy is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy.
Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists.
Matthew Dean Foreman is an American mathematician at University of California, Irvine. He has made notable contributions in set theory and in ergodic theory.
In mathematics, in the field of topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set is perfect if , where denotes the set of all limit points of , also known as the derived set of .
In set theory, Ω-logic is an infinitary logic and deductive system proposed by W. Hugh Woodin (1999) as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure . Just as the axiom of projective determinacy yields a canonical theory of , he sought to find axioms that would give a canonical theory for the larger structure. The theory he developed involves a controversial argument that the continuum hypothesis is false.
In descriptive set theory, the Borel determinacy theorem states that any Gale–Stewart game whose payoff set is a Borel set is determined, meaning that one of the two players will have a winning strategy for the game.
This is a glossary of set theory.
In mathematics, a selection principle is a rule asserting the possibility of obtaining mathematically significant objects by selecting elements from given sequences of sets. The theory of selection principles studies these principles and their relations to other mathematical properties. Selection principles mainly describe covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially function spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property.
