Abstract
In this chapter, we first introduce terminology and notation, then list background results. The reader may skip this chapter and read the necessary parts later when needed. Several results might be learned in graduate courses, but others are advanced and special.
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Notes
- 1.
Their plurals are compacta and continua, respectively.
- 2.
The (topological) dimension of M is equal to n.
- 3.
I.e., the finest (or largest) topology such that each inclusion X γ ⊂ X is continuous. (The term “weak topology” is used with a different meaning by functional analysts, etc.)
- 4.
The wedge (sum) is used for a family of pointed spaces with the common base point.
- 5.
In the case where Y is bounded or X is compact, we can employ the definition
. But, in general, the case d(f, g) = ∞ might occur for this definition. - 6.
A Banach space is a complete normed linear space.
- 7.
Refer to p. 88.
- 8.
The linear span of B is the linear subspace generated by a set B.
- 9.
In some literature, this space is denoted by m( Γ).
- 10.
In some literature, this space is denoted by m.
- 11.
The triangle inequality for ∥x∥p is known as the Minkowski inequality. The proof can be found on pp. 16–17 in [GAGT].
- 12.
See Footnote 5 (p. 8).
- 13.
Recall that spaces are assumed to be Hausdorff.
- 14.
Here, we only consider linear spaces over \(\mathbb {R}\).
- 15.
A flat is also called an affine set, a linear manifold, or a linear variety.
- 16.
The flat hull is also called the affine hull.
- 17.
The core is defined for any subset of F, but we assume that C is convex.
- 18.
Recall that topological spaces are assumed to be Hausdorff. For topological linear spaces (more generally for topological groups), it suffices to assume the axiom T 0, which implies the regularity (cf. Proposition 2.4.2 in [GAGT]).
- 19.
In the Köthe book [(21)], it is called an (F)-norm.
- 20.
As mentioned on p. 112 of [GAGT], (F 5) is unnecessary because it comes from (F 3) and (F 4). But we follow the definition in Köthe’s book [(21)].
- 21.
Proposition 4.1.8 of [GAGT] does not mention about \(\operatorname {rint} C\), but this last equality is easily verified.
- 22.
The plural is polyhedra.
- 23.
Cf. Footnote 3 (p. 5).
- 24.
The plural is subpolyhedra.
- 25.
Cf. Notes for Chapter 4 (p. 247) in [GAGT].
- 26.
Note that K and L are collections of simplexes. This notation f : K → L means a simplicial map f : |K|→|L| with respect to K and L, but it does not mean that f itself is a function between collections. Of course, a simplicial map f : |K|→|L| with respect to K and L induces a function from K to L.
- 27.
In fact, this definition is employed in the textbook of C.P. Rourke and B.J. Sanderson [(17)], p. 16.
- 28.
Here, η is a function between collections K and L but not a map from |K| to |L|.
- 29.
This German terminology was introduced in 1921 by Kneser, but Poincaré claimed this toward the end of nineteenth century and it was formulated as a conjecture in 1908 by Steiniz and Tieze. Refer to Introduction of [113].
- 30.
The definitions of a collar and a collared set are given on p. 129. A PL collar is a closed collar, where a closed collar is defined on p. 135. It is known that every collared closed set in a paracompact space has a closed collar (Proposition 2.5.5). For a bi-collar and a bi-collared set, refer to Remark 2.7.
- 31.
A deformation retract will be defined on p. 249. These concepts will be discussed in Sect. 1.13.
- 32.
See Footnote 30 (p. 185).
- 33.
The simplicial mapping cylinder is dependent on such an order of K (0). See Fig. 4.15 (p. 217) of [GAGT].
- 34.
The proper case of Theorem 1.10.3 is not mentioned in Theorem 4.11.3 of [GAGT] but the same proof is valid.
- 35.
Such a map r is called a retraction, which is discussed in Sect. 1.13.
- 36.
The proof is the same as the Invariance of Domain in [GAGT, p. E2].
- 37.
In many articles, the infinite-dimensionality is not assumed, i.e., w.i.d. = not s.i.d., so f.d. implies w.i.d. However, here we assume the infinite-dimensionality because we discuss the difference among infinite-dimensional spaces.
- 38.
A strong deformation retract was defined on p. 185.
- 39.
Refer to Erratum of [GAGT], Comment for p. 348.
- 40.
It is also called an ∞-equivalence because it is an n-equivalence for every \(n \in \mathbb {N}\).
- 41.
More generally, it is known that every semi-locally contractible topological group is LEC (cf. [GAGT, p. 349]).
- 42.
More generally, this is valid for a space X such that X 2 is normal.
- 43.
It is said that X is unified locally contractile (ULC) if each neighborhood U of ΔX in X 2 contains a neighborhood V of ΔX with a homotopy h : V ×I → U such that h 0 = id, h 1(V ) = ΔX, and pr1h t = pr1|V for every t ∈I. In Theorem 6.3.6 in [GAGT], it is proved that a paracompact space X is ULC if and only if each open cover of X has an h-refinement.
- 44.
- 45.
This corollary does not appear in [GAGT].
. But, in general, the case d(f, g) = ∞ might occur for this definition.References
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Sakai, K. (2020). Preliminaries and Background Results. In: Topology of Infinite-Dimensional Manifolds. Springer Monographs in Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-7575-4_1
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