![arXiv:math/0412558v2 [math.GN] 10 Apr 2016](http://s1.studyres.com/store/data/000780140_1-7e82978ab888d179066aebd70a132571-300x300.png)
arXiv:math/0412558v2 [math.GN] 10 Apr 2016
... perfect space need have countable tightness has never been properly considered. This may be partly because the cardinal invariant Ψ associated with the perfect property is rarely used (it does not even have a name; see page 23). A pertinent example was found by Mr. J. Lo, and it appears as Example 6 ...
... perfect space need have countable tightness has never been properly considered. This may be partly because the cardinal invariant Ψ associated with the perfect property is rarely used (it does not even have a name; see page 23). A pertinent example was found by Mr. J. Lo, and it appears as Example 6 ...
“Research Note” TOPOLOGICAL RING
... A topological groupoid is a groupoid R such that the sets R and R0 are topological spaces, and source, target, object, inverse and composition maps are continuous. Let R and H be two topological groupoids. A morphism of topological groupoids is a pair of maps f:H→R and f0:H0→R0 such that f and f0 ar ...
... A topological groupoid is a groupoid R such that the sets R and R0 are topological spaces, and source, target, object, inverse and composition maps are continuous. Let R and H be two topological groupoids. A morphism of topological groupoids is a pair of maps f:H→R and f0:H0→R0 such that f and f0 ar ...
Axioms of Incidence Geometry Incidence Axiom 1. There exist at
... Lemma 3.3 (Ruler Sliding Lemma). Suppose ` is a line and f W ` ! R is a coordinate function for `. Given a real number c, define a new function f1 W ` ! R by f1 .X/ D f .X/ C c for all X 2 `. Then f1 is also a coordinate function for `. Lemma 3.4 (Ruler Flipping Lemma). Suppose ` is a line and f W ` ...
... Lemma 3.3 (Ruler Sliding Lemma). Suppose ` is a line and f W ` ! R is a coordinate function for `. Given a real number c, define a new function f1 W ` ! R by f1 .X/ D f .X/ C c for all X 2 `. Then f1 is also a coordinate function for `. Lemma 3.4 (Ruler Flipping Lemma). Suppose ` is a line and f W ` ...
General Topology
... Metric spaces do not live in isolation. We can also talk about functions (also called maps or mappings) between them. Typically, we are only interested in the continuous functions. Definition A1.13 Let X and Y be metric spaces. A function f : X → Y is continuous if for all x ∈ X, for all ε > 0, ther ...
... Metric spaces do not live in isolation. We can also talk about functions (also called maps or mappings) between them. Typically, we are only interested in the continuous functions. Definition A1.13 Let X and Y be metric spaces. A function f : X → Y is continuous if for all x ∈ X, for all ε > 0, ther ...
Key Concepts, continued
... that can be proven true by given, definitions, postulates, or already proven theorems •Postulate: a statement that describes a fundamental relationship between basic terms of geometry. Postulates are accepted as true without proof. •Conjecture: an educated guess based on known information 1.8.1: Pro ...
... that can be proven true by given, definitions, postulates, or already proven theorems •Postulate: a statement that describes a fundamental relationship between basic terms of geometry. Postulates are accepted as true without proof. •Conjecture: an educated guess based on known information 1.8.1: Pro ...
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.