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TOPOLOGY 2 - HOMEWORK 1 (1) Prove the following result
... f ∗ is bijective (ie, one-to-one and onto) then f ∗ is a homeomorphism. (Hint: For the second part, prove and then use the following fact: any continuous map from a compact space X to a Hausdorff space Y takes closed subsets of X to closed subsets of Y .) (2) Prove that S 1 × S 1 is homeomorphic to ...
... f ∗ is bijective (ie, one-to-one and onto) then f ∗ is a homeomorphism. (Hint: For the second part, prove and then use the following fact: any continuous map from a compact space X to a Hausdorff space Y takes closed subsets of X to closed subsets of Y .) (2) Prove that S 1 × S 1 is homeomorphic to ...
Algebraic approach to p-local structure of a finite group: Definition 1
... (I) If P ≤ S is fully normalized in F, then P is also fully centralized and AutS (P ) is a Sylow subgroup of (AutF (P )). (II) If P ≤ S and ϕ ∈ HomF (P, S) are such that ϕP is fully centralized, then ϕ extends to ϕ̄ ∈ HomF (Nϕ, S), where Nϕ = {g ∈ NS (P ) | ϕ ◦ cg ◦ ϕ−1 ∈ AutS (ϕP )}. Condition (I) ...
... (I) If P ≤ S is fully normalized in F, then P is also fully centralized and AutS (P ) is a Sylow subgroup of (AutF (P )). (II) If P ≤ S and ϕ ∈ HomF (P, S) are such that ϕP is fully centralized, then ϕ extends to ϕ̄ ∈ HomF (Nϕ, S), where Nϕ = {g ∈ NS (P ) | ϕ ◦ cg ◦ ϕ−1 ∈ AutS (ϕP )}. Condition (I) ...
Pages 1-8
... (8) Show that a morphism φ : X → Y of affine varieties is an isomorphism (i.e. has a two-sided inverse) if and only if φ∗ : k[Y ] → k[X] is an isomorphism of algebras. (9) Suppose that k is of characteristic p > 0. Consider the map φ : An → An , (v1 , . . . , vn ) '→ (v1p , . . . , vnp ). Show that ...
... (8) Show that a morphism φ : X → Y of affine varieties is an isomorphism (i.e. has a two-sided inverse) if and only if φ∗ : k[Y ] → k[X] is an isomorphism of algebras. (9) Suppose that k is of characteristic p > 0. Consider the map φ : An → An , (v1 , . . . , vn ) '→ (v1p , . . . , vnp ). Show that ...
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... More generally, a neighborhood of any subset S of X is defined to be an open set of X containing S. Some authors use the word neighborhood to denote any subset U that contains an open subset containing x. This alternative usage has the advantage that it is easier to develop the theory of filters for ...
... More generally, a neighborhood of any subset S of X is defined to be an open set of X containing S. Some authors use the word neighborhood to denote any subset U that contains an open subset containing x. This alternative usage has the advantage that it is easier to develop the theory of filters for ...
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... A topological space X is said to be paracompact if every open cover of X has a locally finite open refinement. In more detail, if (Ui )i∈I is any family of open subsets of X such that ∪i∈I Ui = X , then there exists another family (Vi )i∈I of open sets such that ∪i∈I Vi = X Vi ⊂ Ui for all i ∈ I and ...
... A topological space X is said to be paracompact if every open cover of X has a locally finite open refinement. In more detail, if (Ui )i∈I is any family of open subsets of X such that ∪i∈I Ui = X , then there exists another family (Vi )i∈I of open sets such that ∪i∈I Vi = X Vi ⊂ Ui for all i ∈ I and ...