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Trivial topology; Cofinite topology; Finer topology; Product topology. Restricted product; Quotient space; Unit interval; Continuum (topology) Extended real number line; Long line (topology) Sierpinski space; Cantor set, Cantor space, Cantor cube; Space-filling curve; Topologist's sine curve; Uniform norm; Weak topology; Strong topology ...
The overlapping interval topology is a simple example of a topology that is T 0 but is not T 1. Every weakly Hausdorff space is T 1 but the converse is not true in general. The cofinite topology on an infinite set is a simple example of a topology that is T 1 but is not Hausdorff (T 2). This follows since no two nonempty open sets of the ...
If is a set equipped with a mapping satisfying the above properties, then the set of all possible outputs of int satisfies the previous axioms for open sets, and hence defines a topology; it is the unique topology whose associated interior operator coincides with the given int. [28] It follows that on a topological space , all definitions can ...
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
Let τ 1 and τ 2 be two topologies on a set X and let B i (x) be a local base for the topology τ i at x ∈ X for i = 1,2. Then τ 1 ⊆ τ 2 if and only if for all x ∈ X, each open set U 1 in B 1 (x) contains some open set U 2 in B 2 (x). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.
In topology, the pasting or gluing lemma, and sometimes the gluing rule, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions .
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds , with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak ...