Banach Algebras
... We have seen that if A is without identity we can adjoin an e ∈ A[e]. But how does this help? If x ∈ A it is still not invertible in A[e]. And if A does have it’s own identity, say, e0 , then invertibility in A and in A[e] have little to do with each other. As a matter of fact, so far we haven’t at ...
... We have seen that if A is without identity we can adjoin an e ∈ A[e]. But how does this help? If x ∈ A it is still not invertible in A[e]. And if A does have it’s own identity, say, e0 , then invertibility in A and in A[e] have little to do with each other. As a matter of fact, so far we haven’t at ...
On Noetherian Spaces - LSV
... topology on spectra of Noetherian rings was the first known Noetherian topology; we discuss it only in Section 8, in the light of the rest of our paper. Our contribution occupies the other sections. We first show the tight relationship between well-quasi orders and Noetherian spaces in Section 3, an ...
... topology on spectra of Noetherian rings was the first known Noetherian topology; we discuss it only in Section 8, in the light of the rest of our paper. Our contribution occupies the other sections. We first show the tight relationship between well-quasi orders and Noetherian spaces in Section 3, an ...
METRIC SPACES
... 1.4.1. Open sets. Let (X, d) be a metric space. Definition 1.25. Let A ⊂ X be a subset. (1) A point x ∈ A is called an internal (or interior) point if there exists r > 0 such that B(x, r) ⊂ A. (2) A subset A ⊂ X is called an open set if every point of A is an internal point. That is, if for any x ∈ ...
... 1.4.1. Open sets. Let (X, d) be a metric space. Definition 1.25. Let A ⊂ X be a subset. (1) A point x ∈ A is called an internal (or interior) point if there exists r > 0 such that B(x, r) ⊂ A. (2) A subset A ⊂ X is called an open set if every point of A is an internal point. That is, if for any x ∈ ...
r*bg* -Closed Sets in Topological Spaces.
... Proof: Let A be r*bg*- closed set in X. Let U be regular open set such that A⊆U.Since every regular open set is b- open and A is r*bg*- closed set, we have cl(A) ⊆ rbcl(A) ⊆ U.Therefore cl(A) ⊆ U.Hence A is rg closed set in X. The converse of the above theorem need not be true as seen from the follo ...
... Proof: Let A be r*bg*- closed set in X. Let U be regular open set such that A⊆U.Since every regular open set is b- open and A is r*bg*- closed set, we have cl(A) ⊆ rbcl(A) ⊆ U.Therefore cl(A) ⊆ U.Hence A is rg closed set in X. The converse of the above theorem need not be true as seen from the follo ...
local contractibility, cell-like maps, and dimension
... Example [7] to get a cell-like map /: 7°° -» X such that X is not movable. In particular, X cannot have the shape of an AR and is therefore not contractible. By Lemma 2.1 there consequently exists a cell-like image of 7°° such that no nonempty open subset is contractible in the space. Alternatively, ...
... Example [7] to get a cell-like map /: 7°° -» X such that X is not movable. In particular, X cannot have the shape of an AR and is therefore not contractible. By Lemma 2.1 there consequently exists a cell-like image of 7°° such that no nonempty open subset is contractible in the space. Alternatively, ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.