BASIC ALGEBRAIC TOPOLOGY: THE FUNDAMENTAL GROUP OF
... Example 1.4. In R , any two paths f0 and f1 that have the same endpoints are homotopic via the linear homotopoy defined by ft (s) = (1 − t)f0 (s) + tf1 (s). This means that each f0 (s) travels along the line segments to f1 (s) at a constant speed. Proposition 1.5. Given a topological space X with tw ...
... Example 1.4. In R , any two paths f0 and f1 that have the same endpoints are homotopic via the linear homotopoy defined by ft (s) = (1 − t)f0 (s) + tf1 (s). This means that each f0 (s) travels along the line segments to f1 (s) at a constant speed. Proposition 1.5. Given a topological space X with tw ...
Mathematical Preliminaries
... not within a set, where links are unique. This is the intuitive definition of a map. There are four types of maps, which describe the nature of the links between the two sets. A map is one-to-one if every element in the domain has no more than one link. A map is manyto-one if the domain has the prop ...
... not within a set, where links are unique. This is the intuitive definition of a map. There are four types of maps, which describe the nature of the links between the two sets. A map is one-to-one if every element in the domain has no more than one link. A map is manyto-one if the domain has the prop ...
ON (g, s)-CONTINUOUS AND (πg, s)
... The finite union of regular open sets is said to be π-open [42]. The complement of π-open set is said to be π-closed. A subset A of a space X is said to be generalized closed (briefly, g-closed) [24] (resp. πg-closed [12]) if cl(A) ⊂ U whenever A ⊂ U and U is open (resp. π-open) in X. If the complem ...
... The finite union of regular open sets is said to be π-open [42]. The complement of π-open set is said to be π-closed. A subset A of a space X is said to be generalized closed (briefly, g-closed) [24] (resp. πg-closed [12]) if cl(A) ⊂ U whenever A ⊂ U and U is open (resp. π-open) in X. If the complem ...
on some very strong compactness conditions
... each proper closed subspace is finite and thus strongly S-closed. By Theorem 2.6, (X, τ ) is LC-compact but fails to be λC-compact by Theorem 2.7. Example 3.5 (another example of an LC-compact space). Let τ1 be the cofinite topology on X and let τ2 be the point-generated topology on X with respect to ...
... each proper closed subspace is finite and thus strongly S-closed. By Theorem 2.6, (X, τ ) is LC-compact but fails to be λC-compact by Theorem 2.7. Example 3.5 (another example of an LC-compact space). Let τ1 be the cofinite topology on X and let τ2 be the point-generated topology on X with respect to ...
Chapter 5
... 2. ∀ (x ∈ U ⊂ X with U open), ∃ (f ∈ A positive, supported in U , with f (x) > 0 ). Secondly, 2nd countability and local compactness imply paracompactness: Theorem 5.20. Any Hausdorff, locally compact and 2nd countable space is paracompact. Proof. (of Theorem 5.20) We use an exhaustion {Kn } of X (T ...
... 2. ∀ (x ∈ U ⊂ X with U open), ∃ (f ∈ A positive, supported in U , with f (x) > 0 ). Secondly, 2nd countability and local compactness imply paracompactness: Theorem 5.20. Any Hausdorff, locally compact and 2nd countable space is paracompact. Proof. (of Theorem 5.20) We use an exhaustion {Kn } of X (T ...
On Colimits in Various Categories of Manifolds
... There is one subtlety which the proof above leaves out. We computed the pushout in the category of topological spaces and saw that it’s not a manifold. But how do we know that the pushout in the category of manifolds is the same as the pushout in the category of topological spaces? This is addressed ...
... There is one subtlety which the proof above leaves out. We computed the pushout in the category of topological spaces and saw that it’s not a manifold. But how do we know that the pushout in the category of manifolds is the same as the pushout in the category of topological spaces? This is addressed ...