scholarly journals Cohomology Relations in Spaces with a Topological Transformation Group

1953 ◽  
Vol 5 ◽  
pp. 113-125
Author(s):  
Sze-Tsen hu
Keyword(s):  
Topological Space ◽  
Transformation Group ◽  
Orbit Space ◽  
Topological Transformation ◽  
Continuous Map ◽  
Open Map

Let Q be a topological transformation group operating on the left of a topological space X. Let us denote by. B the orbit space and p : X → B the projection, p is a continuous and open map of X onto B. For an arbitrary abelian coefficient group G, the continuous map p induces homomorphisms

Symmetric Powers of Motives

2019 ◽  
pp. 232-252
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Topological Space ◽  
Symmetric Group ◽  
Homotopy Group ◽  
Orbit Space ◽  
Symmetric Power ◽  
Homotopy Groups ◽  
Integral Homology ◽  
Symmetric Powers ◽  
Mac Lane ◽  
Abelian Monoid

This chapter develops the basic theory of symmetric powers of smooth varieties. The constructions in this chapter are based on an analogy with the corresponding symmetric power constructions in topology. If 𝐾 is a set (or even a topological space) then the symmetric power 𝑆𝑚𝐾 is defined to be the orbit space 𝐾𝑚/Σ‎𝑚, where Σ‎𝑚 is the symmetric group. If 𝐾 is pointed, there is an inclusion 𝑆𝑚𝐾 ⊂ 𝑆𝑚+1𝐾 and 𝑆∞𝐾 = ∪𝑆𝑚𝐾 is the free abelian monoid on 𝐾 − {*}. When 𝐾 is a connected topological space, the Dold–Thom theorem says that ̃𝐻*(𝐾, ℤ) agrees with the homotopy groups π‎ *(𝑆∞𝐾). In particular, the spaces 𝑆∞(𝑆 𝑛) have only one homotopy group (𝑛 ≥ 1) and hence are the Eilenberg–Mac Lane spaces 𝐾(ℤ, 𝑛) which classify integral homology.

On Certain Onto Maps

1962 ◽  
Vol 14 ◽  
pp. 461-466 ◽  
Author(s):  
Isaac Namioka
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Dimensional Space ◽  
Extension Theorem ◽  
Normal Space ◽  
Continuous Map ◽  
Image Position

Let Δn (n > 0) denote the subset of the Euclidean (n + 1)-dimensional space defined byA subset σ of Δn is called a face if there exists a sequence 0 ≤ i1 ≤ i2 ≤ … < im ≤ n such thatand the dimension of σ is defined to be (n — m). Let denote the union of all faces of Δn of dimensions less than n. A topological space Y is called solid if any continuous map on a closed subspace A of a normal space X into Y can be extended to a map on X into Y. By Tietz's extension theorem, each face of Δn is solid. The present paper is concerned with a generalization of the following theorem which seems well known.

Zero-Dimensional Compactifications of Locally Compact Spaces

1974 ◽  
Vol 26 (4) ◽  
pp. 920-930 ◽  
Author(s):  
R. Grant Woods
Keyword(s):  
Topological Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dense Subspace ◽  
Locally Compact ◽  
Partially Order ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Continuous Map ◽  
Locally Compact Spaces

Let X be a locally compact Hausdorff topological space. A compactification of X is a compact Hausdorff space which contains X as a dense subspace. Two compactifications αX and γX of X are equivalent if there is a homeomorphism from αX onto γX that fixes X pointwise. We shall identify equivalent compactifications of a given space. If is a family of compactifications of X, we can partially order by saying that αX ≦ γX if there is a continuous map from γX onto αX that fixes X pointwise.

Local Topological Properties of Maps and Open Extensions of Maps

1977 ◽  
Vol 29 (6) ◽  
pp. 1121-1128
Author(s):  
J. K. Kohli
Keyword(s):  
Topological Space ◽  
Open Subset ◽  
Countable Union ◽  
Dense Open Subset ◽  
Topological Properties ◽  
Hausdorff Spaces ◽  
Local Homeomorphism ◽  
Open Map

A σ-discrete set in a topological space is a set which is a countable union of discrete closed subsets. A mapping ƒ : X ⟶ Y from a topological space X into a topological space Y is said to be σ-discrete (countable) if each fibre ƒ-1(y), y ϵ Y is σ-discrete (countable). In 1936, Alexandroff showed that every open map of a bounded multiplicity between Hausdorff spaces is a local homeomorphism on a dense open subset of the domain [2].

Dimension of a Topological Transformation Group

1976 ◽  
Vol 28 (3) ◽  
pp. 594-599
Author(s):  
Hsu-Tung Ku ◽  
Mei-Chin Ku
Keyword(s):  
Lie Group ◽  
Effective Action ◽  
Transformation Group ◽  
Topological Transformation ◽  
Connected Lie Group

Throughout this paper, the Alexander-Spanier cohomology with compact supports will be used. Suppose X is a compact connected topological ra-manifold which admits an effective action of a compact connected Lie group G (m ≧ 19).

On the Irregular Sets of a Transformation Group

1973 ◽  
Vol 16 (2) ◽  
pp. 225-232 ◽  
Author(s):  
S. K. Kaul
Keyword(s):  
Topological Space ◽  
Transformation Group ◽  
Open Set ◽  
Group 2 ◽  
Set Of Points

We assume throughout that (X, T, π) is a transformation group [2], where X is a topological space which is always assumed to be regular and Hausdorff. We call a point x ∊ X regular under T if for any open set U in X and any subset G of T such that , there exists an open set V containing x, such that VG ⊆ U [7]. Let R(X) denote the interior of the set of all the regular points of X under T, and I(X) the set of irregular points of X under T, that is the set of points which are not regular under T.

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