as a PDF
... of Day and Kelly where they proved that the functor − × Y preserves quotient maps if and only if the lattice O(Y ) of open subsets of Y is a continuous lattice in the sense of Scott [S], and that a Hausdorff space satisfies this property if and only if it is locally compact. Since − × Y preserves co ...
... of Day and Kelly where they proved that the functor − × Y preserves quotient maps if and only if the lattice O(Y ) of open subsets of Y is a continuous lattice in the sense of Scott [S], and that a Hausdorff space satisfies this property if and only if it is locally compact. Since − × Y preserves co ...
this PDF file - European Journal of Pure and Applied
... and (A ∩ B) \ cl(G ∩ H) = (A ∩ B) \ X = ; ∈ I, it follows that A ∩ B is I − β-open. Remark 2. In Example 2, we saw that the singleton subset {b} was not β-open with respect to the minimal ideal I = {;}. Notice that the set {a, b} = {a} ∪ {b} is β-open with respect to I = {;}, simply because the non- ...
... and (A ∩ B) \ cl(G ∩ H) = (A ∩ B) \ X = ; ∈ I, it follows that A ∩ B is I − β-open. Remark 2. In Example 2, we saw that the singleton subset {b} was not β-open with respect to the minimal ideal I = {;}. Notice that the set {a, b} = {a} ∪ {b} is β-open with respect to I = {;}, simply because the non- ...
The inverse image of a metric space under a biquotient
... st(x, ri) (X-L(U)) ~ ~ for each i. Then by the hypothesis L(x) n cl(L(X-L(U))) ~ 0. Since it is easy to see that L(X-L(U)) X-L(U), therefore L(x) n cl(X-L(U)) ~ 0. But L(x) n cl(X-L(U))~~ implies every neighborhood of L(x) has a nonempty intersection with X-L(U), which is not true as U is a neighbor ...
... st(x, ri) (X-L(U)) ~ ~ for each i. Then by the hypothesis L(x) n cl(L(X-L(U))) ~ 0. Since it is easy to see that L(X-L(U)) X-L(U), therefore L(x) n cl(X-L(U)) ~ 0. But L(x) n cl(X-L(U))~~ implies every neighborhood of L(x) has a nonempty intersection with X-L(U), which is not true as U is a neighbor ...
![arXiv:math/9811003v1 [math.GN] 1 Nov 1998](http://s1.studyres.com/store/data/000335093_1-58b56f331c623a9d53167347869d2e0a-300x300.png)
Exponential laws for topological categories, groupoids
... the space M (D, E) , where E is a k-group and D is a colimit of k-groups. The use of k-groupoids rather than just k-groups, however, is not orrly required by our method of proof, but has the advantage of easily giving results on free k-groups. By generalising still further to k-categories i+e obtain ...
... the space M (D, E) , where E is a k-group and D is a colimit of k-groups. The use of k-groupoids rather than just k-groups, however, is not orrly required by our method of proof, but has the advantage of easily giving results on free k-groups. By generalising still further to k-categories i+e obtain ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.