CW complexes
... The boundary of Dn in Rn is the standard (n − 1)-sphere S n−1 = { x ∈ Rn : |x| = 1 }. We note that the 0-disk D0 is equal to R0 = {0} by definition. We have int(D0 ) = D0 = {0}. A topological space X is called quasi-compact if every open cover of X has a finite subcover, i.e. wheneverP {Ui }i∈I is a ...
... The boundary of Dn in Rn is the standard (n − 1)-sphere S n−1 = { x ∈ Rn : |x| = 1 }. We note that the 0-disk D0 is equal to R0 = {0} by definition. We have int(D0 ) = D0 = {0}. A topological space X is called quasi-compact if every open cover of X has a finite subcover, i.e. wheneverP {Ui }i∈I is a ...
PROOF. Let a = ∫X f dµ/µ(X). By convexity the graph of g lies
... a homeomorphism h : U → B ⊂ Rn is called a chart or a system of local coordinates. A topological manifold with boundary is a Hausdorff space X with a countable base for the topology such that every point is contained in an open set homeomorphic to an open set in Rn−1 × [0, ∞). R EMARK A.6.24. One ea ...
... a homeomorphism h : U → B ⊂ Rn is called a chart or a system of local coordinates. A topological manifold with boundary is a Hausdorff space X with a countable base for the topology such that every point is contained in an open set homeomorphic to an open set in Rn−1 × [0, ∞). R EMARK A.6.24. One ea ...
RADON-NIKOD´YM COMPACT SPACES OF LOW WEIGHT AND
... The main result in section 1 is the following: Theorem 3. If K is a quasi Radon-Nikodým compact space of weight less than b, then K is Radon-Nikodým compact. The weight of a topological space is the least cardinality of a base for its topology. We also recall the definition of cardinal b. In the s ...
... The main result in section 1 is the following: Theorem 3. If K is a quasi Radon-Nikodým compact space of weight less than b, then K is Radon-Nikodým compact. The weight of a topological space is the least cardinality of a base for its topology. We also recall the definition of cardinal b. In the s ...
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.