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A collection of subsets of a topological space is called σ-locally finite [6] [7] or countably locally finite [8] if it is a countable union of locally finite collections. The σ-locally finite notion is a key ingredient in the Nagata–Smirnov metrization theorem , which states that a topological space is metrizable if and only if it is ...
A bijection between two topological spaces is a homeomorphism if and only if the derived set of the image (in the second space) of any subset of the first space is the image of the derived set of that subset. [7] A space is a T 1 space if every subset consisting of a single point is closed. [8]
For instance, had been declared as a subset of , with the sets and not necessarily related to each other in any way, then would likely mean instead of . If it is needed then unless indicated otherwise, it should be assumed that X {\displaystyle X} denotes the universe set , which means that all sets that are used in the formula are subsets of X ...
In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).
The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
A partition of a set is defined as a family of nonempty, pairwise disjoint subsets of whose union is . For example, B 3 = 5 {\displaystyle B_{3}=5} because the 3-element set { a , b , c } {\displaystyle \{a,b,c\}} can be partitioned in 5 distinct ways:
(Proposition (9.9.4)) One important role that these constructibility results have is that in most cases assuming the morphisms in questions are also flat it follows that the properties in question in fact hold in an open subset. A substantial number of such results is included in EGA IV § 12. [11]
Let be a set and a nonempty family of subsets of ; that is, is a nonempty subset of the power set of . Then is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong finite intersection property if that intersection is always infinite.