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The weak topology of a CW complex is defined as a direct limit. Let X {\displaystyle X} be any directed set with a greatest element m {\displaystyle m} . The direct limit of any corresponding direct system is isomorphic to X m {\displaystyle X_{m}} and the canonical morphism ϕ m : X m → X {\displaystyle \phi _{m}:X_{m}\rightarrow X} is an ...
In mathematics, a limit is the value that a function (or sequence) ... In topology, limits are used to define limit points of a subset of a topological space, ...
In mathematics, especially in algebraic topology, the homotopy limit and colimit [1] pg 52 are variants of the notions of limit and colimit extended to the homotopy category (). The main idea is this: if we have a diagram
A limit point can be characterized as an adherent point that is not an isolated point. Limit points of a set should also not be confused with boundary points. For example, is a boundary point (but not a limit point) of the set {} in with standard topology.
Virtually all concepts of topology can be rephrased in the language of nets and limits. This may be useful to guide the intuition since the notion of limit of a net is very similar to that of limit of a sequence. The following set of theorems and lemmas help cement that similarity:
The Sorgenfrey line can thus be used to study right-sided limits: if : is a function, then the ordinary right-sided limit of at (when the codomain carries the standard topology) is the same as the usual limit of at when the domain is equipped with the lower limit topology and the codomain carries the standard topology.
This is known as the limit topology. The set of infinite strings is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. As the original spaces are discrete , the limit space is totally disconnected .
The limit of F is called an inverse limit or projective limit. If J = 1, the category with a single object and morphism, then a diagram of shape J is essentially just an object X of C. A cone to an object X is just a morphism with codomain X. A morphism f : Y → X is a limit of the diagram X if and only if f is an isomorphism.