In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply...

denoted by SX or susp(X).: 76  There is a variation of the suspension for pointed space, which is called the reduced suspension and denoted by ΣX. The "usual"...

category of pointed spaces is obtained from the category Top* by inverting the pointed maps that are weak homotopy equivalences. For pointed spaces X and Y...

Basepoint may refer to a point singled out in a: Pointed set, or in a Pointed space Origin (mathematics) This disambiguation page lists mathematics articles...

constraints. Often, one works with a pointed space—that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserves...

functor from groups to pointed sets.: 582  A pointed set may be seen as a pointed space under the discrete topology or as a vector space over the field with...

In algebraic topology, the path space fibration over a pointed space ( X , ∗ ) {\displaystyle (X,*)} is a fibration of the form Ω X ↪ P X → χ ↦ χ ( 1 )...

A pointed arch, ogival arch, or Gothic arch is an arch with a pointed crown meet at an angle at the top of the arch. Also known as a two-centred arch...

"one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces (i.e. topological spaces with distinguished basepoints x 0 {\displaystyle...

product of two pointed spaces (i.e. topological spaces with distinguished basepoints) (X, x0) and (Y, y0) is the quotient of the product space X × Y under...

vector, an analogous point of a vector space Distance from a point to a plane Pointed space, a topological space with a distinguished point Radial basis...

mathematics, the based path space P X {\displaystyle PX} of a pointed space ( X , ∗ ) {\displaystyle (X,*)} is the space that consists of all maps f {\displaystyle...

out a basepoint. The tangent space construction can be viewed as a functor from Man•p to VectR as follows: given pointed manifolds ( M , p 0 ) {\displaystyle...

mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S1 to...

map between nilpotent spaces is a disjoint union of nilpotent spaces. Moreover, the null component of the pointed mapping space Map ∗ ⁡ ( K , X ) {\displaystyle...

{\displaystyle X\times \{y\}\cup \{x\}\times Y} . A finite wedge of a pointed space ( X , x ) {\displaystyle (X,x)} is denoted Z t r ( X ∧ q ) = Z t r (...

homotopy category of (pointed) spaces to the category of commutative rings. Thus, for instance, the K-theory over contractible spaces is always Z . {\displaystyle...

equivalence classes. If ( X , x 0 ) {\displaystyle (X,x_{0})} is a pointed space, there is a related construction, the reduced cone, given by ( X × [...

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was the Freudenthal suspension theorem, which states that given any pointed space X {\displaystyle X} , the homotopy groups π n + k ( Σ n X ) {\displaystyle...

tangent space, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise, cotangent space is...

space Lindelöf space Sigma-compact space Connected space T0 space T1 space Hausdorff space Completely Hausdorff space Regular space Tychonoff space Normal...

of pointed topological spaces, defined by requiring maps to preserve base points. The articles compactly generated space and weak Hausdorff space define...

of a space X {\displaystyle X} is often denoted π 0 ( X ) . {\displaystyle \pi _{0}(X).} One can also define paths and loops in pointed spaces, which...

false without the restriction to connected pointed spaces, and an analogous statement for unpointed spaces is also false. A similar statement does, however...

non-compact spaces. A pointed metric space is a pair (X,p) consisting of a metric space X and point p in X. A sequence (Xn, pn) of pointed metric spaces converges...

map through maps sending e to e. This may be thought of as a pointed topological space together with a continuous multiplication for which the basepoint...

not taken up to homotopy) in a pointed space X, endowed with the compact open topology, is known as the loop space, denoted Ω X . {\displaystyle \Omega...

pointed spaces. Given a map f : X → Y {\displaystyle f\colon X\to Y} , the mapping cone C f {\displaystyle C_{f}} is defined to be the quotient space...

homotopy excision theorem. Let X be an n-connected pointed space (a pointed CW-complex or pointed simplicial set). The map X → Ω ( Σ X ) {\displaystyle...