topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
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fiber space, space attachment
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closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
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homotopy theory, (∞,1)-category theory, homotopy type theory
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see also algebraic topology
Introductions
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For a pointed topological space $(X, x)$, its reduced suspension $\Sigma (X,x)$ is obtained from the plain suspension
of the underlying topological space $X$ by collapsing the meridian through the basepoint $x$ itself to a point — this making $\Sigma(X,x)$ itself a pointed topological space with basepoint the equivalence class of that meridian $mer(x)$:
(Notice that this identifies in particular also the two antipodal “poles” of the plain suspension.)
If $X$ admits the structure of a CW-complex then, under passage to the classical homotopy category of pointed topological spaces (cf. here) this construction models the homotopy pushout of the terminal map $(X,x) \to (\ast,pt)$ along itself, which explains its prevalence in homotopy theory (especially in stable homotopy theory, see also at suspension spectrum).
Moreover, in this case of CW-complexes the underlying space of $\Sigma (X,x)$ (i.e. forgetting its basepoint) is weakly homotopy equivalent to the plain suspension $\mathrm{S} X$ of the underlying space $X$ of $(X,x)$. In this sense, reduced suspension in the context of homotopy theory may be understood as just being plain suspension but with basepoints taken into account.
For $(X,x)$ a pointed topological space, then its reduced suspension $\Sigma X$ is equivalently the following:
obtained from the standard cylinder $I\times X$ (product topological space with the closed interval $I = [0,1]$) by identifying the subspace $(\{0,1\}\times X) \cup (I\times \{x\})$ to a point, i.e. the quotient space (this example)
(Think of crushing the two ends of the cylinder and the line through the base point to a point.)
obtained from the plain suspension
of $X$ by passing to the quotient space which collapses $\{x\} \times I$ to a point (this example)
For the purposes of generalized (Eilenberg-Steenrod) cohomology theory typoically it does not matter whether one evaluates on the standard suspension or the reduced suspension. For example for topological K-theory since $\{x\} \times I$ is a contractible closed subspace, then this prop. says that topological vector bundles do not see a difference as long as $X$ is a compact Hausdorff space.
obtained from the reduced cylinder by collapsing the two ends, i.e. the cofiber
the mapping cone in pointed topological spaces formed with respect to the reduced cylinder $X \wedge (I_+)$ of the map $X \to \ast$;
the smash product $S^1\wedge X$, of $X$ with the circle (based at some point) with $X$.
For CW-complexes $X$ that are also pointed, with the point identified with a 0-cell, then their reduced suspension is weakly homotopy equivalent to the ordinary suspension: $\Sigma X \simeq S X$.
suspensions are H-cogroup objects
Up to homeomorphism, the reduced suspension of the $n$-sphere is the $(n+1)$-sphere
See at one-point compactification – Examples – Spheres for details.
Discussion of (reduced) suspension may be found in most introductions to homotopy theory (for discussion of unreduced suspension see also there).
For instance:
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, §2.10 in: Algebraic topology from a homotopical viewpoint, Springer (2008) [doi:10.1007/b97586]
Jeffrey Strom, §3.8 and §17 in: Modern classical homotopy theory, Graduate Studies in Mathematics 127, American Mathematical Society (2011) [doi:10.1090/gsm/127]
Martin Arkowitz, Loop spaces and suspensions, §2.3 in: Introduction to Homotopy Theory, Springer (2011) [doi:10.1007/978-1-4419-7329-0]
Review in the context of stable homotopy theory:
Last revised on January 6, 2023 at 14:28:10. See the history of this page for a list of all contributions to it.