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The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.
A Peano continuum is a continuum that is locally connected at each point. An indecomposable continuum is a continuum that cannot be represented as the union of two proper subcontinua. A continuum X is hereditarily indecomposable if every subcontinuum of X is indecomposable. The dimension of a continuum usually means its topological dimension.
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counterintuitive, as the common meanings of open and closed are antonyms, but their mathematical definitions are not mutually exclusive .
Though the subspace topology of Y = {−1} ∪ {1/n } n∈N in the section above is shown not to be generated by the induced order on Y, it is nonetheless an order topology on Y; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open in Y for every y in Y), so the subspace topology is the discrete topology on Y (the topology in which every subset of Y is open ...
Definition: [4] We say that f has a closed graph in X × Y if the graph of f, Gr f, is a closed subset of X × Y when X × Y is endowed with the product topology. If S = X or if X is clear from context then we may omit writing "in X × Y " Note that we may define an open graph, a sequentially closed graph, and a sequentially open graph in ...
Absolutely closed See H-closed Accessible See . Accumulation point See limit point. Alexandrov topology The topology of a space X is an Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of a poset.
Let be the least uncountable ordinal.In an analog of Baire space derived from the -fold cartesian product of with itself, any closed set is the disjoint union of an -perfect set and a set of cardinality, where -closedness of a set is defined via a topological game in which members of are played.
The name open–closed principle has been used in two ways. Both ways use generalizations (for instance, inheritance or delegate functions) to resolve the apparent dilemma, but the goals, techniques, and results are different. The open–closed principle is one of the five SOLID principles of object-oriented design.