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The Topologist's sine curve, a useful example in point-set topology.It is connected but not path-connected. In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology.
For example, is a boundary point (but not a limit point) of the set {} in with standard topology. However, is a limit point (though not a boundary point) of interval [,] in with standard topology (for a less trivial example of a limit point, see the first caption). [3] [4] [5]
In mathematics, more specifically in point-set topology, the derived set of a subset of a topological space is the set of all limit points of . It is usually denoted by S ′ . {\displaystyle S'.} The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line .
A set with no isolated point is said to be dense-in-itself (every neighbourhood of a point contains other points of the set). A closed set with no isolated point is called a perfect set (it contains all its limit points and no isolated points). The number of isolated points is a topological invariant, i.e. if two topological spaces X, Y are ...
The following topologies are a known source of counterexamples for point-set topology. Alexandroff plank; Appert topology − A Hausdorff, perfectly normal (T 6), zero-dimensional space that is countable, but neither first countable, locally compact, nor countably compact. Arens square
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.
General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. [11] [12] 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.
Topos theory is, in some sense, a generalization of classical point-set topology. One should therefore expect to see old and new instances of pathological behavior. For instance, there is an example due to Pierre Deligne of a nontrivial topos that has no points (see below for the definition of points of a topos).