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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 ...
In numerical analysis, given a square grid in two dimensions, the nine-point stencil of a point in the grid is a stencil made up of the point itself together with its eight "neighbors". It is used to write finite difference approximations to derivatives at grid points.
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 ...
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.
If is a subset of a topological space then the limit of a convergent sequence in does not necessarily belong to , however it is always an adherent point of . Let ( x n ) n ∈ N {\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }} be such a sequence and let x {\displaystyle x} be its limit.
Examples of perfect subsets of the real line are the empty set, all closed intervals, the real line itself, and the Cantor set. The latter is noteworthy in that it is totally disconnected . Whether a set is perfect or not (and whether it is closed or not) depends on the surrounding space.
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. [1] [2] In a topological space, a closed set can be defined as a set which contains all its limit points.