quaternionic projective line$\,\mathbb{H}P^1$
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
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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, (β,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directedβ¦
models: topological, simplicial, localic, β¦
see also algebraic topology
Introductions
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Basic facts
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There are several ways to make sense of the notion of the n-sphere in the limit (rather: colimit) that $n \to \infty$. Typically the resulting infinite-dimensional sphere $S^\infty$ has the remarkable property that it is contractible as a topological space β or at least weakly contractible, in that its map to the point is a weak homotopy equivalence.
Due to their weak homotopy equivalence to points, in homotopy theory infinite-dimensional spheres provide nothing new in themselves, but as a source of big contractible spaces they serve as a starting point for making concrete models of classifying spaces.
For example, the 2k-sphere is the total space of the tautological circle group-principal bundle over complex projective k-space, so that in the colimit $2k \to \infty$ the infinite-dimensional sphere emerges as a model for the universal principal bundle over the classifying space $B \mathrm{U}(1)$. This being contractible relates to important statements such as that the zero-section into the Thom space of the universal line bundle is a weak equivalence.
Realizing the n-sphere as a cell complex of sorts, such that $S^n \hookrightarrow S^{n+1}$ is a relative cell complex-inclusion for all $n \in \mathbb{N}$, the infinite-dimensional sphere may be taken to be the colimit over this sequence of inclusions. Since the n+1-sphere is n-connected it follows that $S^\infty$ is $\infty$-connected and hence weakly contractible:
Specifically in the category Top of topological spaces, the n-sphere has a standard CW-complex structure with exactly 2-cells in each dimension, obtained inductively by attaching two $n$-dimensional hemispheres to the $(n-1)$-sphere regarded as the equator in the $n$-sphere.
Since forming homotopy groups $\pi_k(-)$ commutes with taking the colimit over these relative cell complex inclusions (by the skeleton filtration), and since the n-sphere has trivial homotopy groups in dimension $k \lt n$, it follows at once that the homotopy groups of the infinite-dimensional sphere all vanish, and hence that it is weakly equivalent to the point in the classical model structure on topological spaces:
One can also talk about a sphere in an arbitrary (possibly infinite-dimensional) normed vector space $V$:
If a locally convex topological vector space admits a continuous linear injection into a normed vector space, this can be used to define its sphere. If not, one can still define the sphere as a quotient of the space of non-zero vectors under the scalar action of $(0,\infty)$.
Homotopy theorists define $S^\infty$ to be the sphere in the (incomplete) normed vector space (traditionally with the $l^2$ norm) of infinite sequences almost all of whose values are $0$, which is the directed colimit of the $S^n$:
If the vector space is a shift space, then contractibility is straightforward to prove.
Let $V$ be a shift space of some order. Let $S V$ be its sphere (either via a norm or as the quotient of non-zero vectors). Then $S V$ is contractible.
Let $T \colon V \to V$ be a shift map. The idea is to homotop the sphere onto the image of $T$, and then down to a point.
It is simplest to start with the non-zero vectors, $V \setminus \{0\}$. As $T$ is injective, it restricts to a map from this space to itself which commutes with the scalar action of $(0,\infty)$. Define a homotopy $H \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}$ by $H_t(v) = (1 - t)v + t T v$. It is clear that, assuming it is well-defined, it is a homotopy from the identity to $T$. To see that it is well-defined, we need to show that $H_t(v)$ is never zero. The only place where it could be zero would be on an eigenvector of $T$, but as $T$ is a shift map then it has none.
As $T$ is a shift map, it is not surjective and so we can pick some $v_0$ not in its image. Then we define a homotopy $G \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}$ by $G_t(v) = (1 - t)T v + t v_0$. As $v_0$ is not in the image of $T$, this is well-defined on $V \setminus \{0\}$. Combining these two homotopies results in the desired contraction of $V \setminus \{0\}$.
If $V$ admits a suitable function defining a spherical subset (such as a norm) then we can modify the above to a contraction of the spherical subset simply by dividing out by this function. If not, as the homotopies above all commute with the scalar action of $(0,\infty)$, they descend to the definition of the sphere as the quotient of $V \setminus \{0\}$.
Last revised on March 5, 2024 at 00:05:44. See the history of this page for a list of all contributions to it.