WHAT IS A TOPOLOGICAL STACK? 1. introduction Stacks were
... orbits. For instance, [X/G] in some sense remembers all the stabilizer groups of the action, while X/G is completely blind to them. There is a natural morphism πmod : [X/G] → X/G enabling us to compare the stacky quotient [X/G] with the coarse quotient X/G. We will encounter πmod again later in this ...
... orbits. For instance, [X/G] in some sense remembers all the stabilizer groups of the action, while X/G is completely blind to them. There is a natural morphism πmod : [X/G] → X/G enabling us to compare the stacky quotient [X/G] with the coarse quotient X/G. We will encounter πmod again later in this ...
Quotient spaces
... Quotient spaces Up to this point in the text, when introducing new concepts for topological spaces such as Hausdorffness, compactness, connectedness etc, we have relied almost exclusively on the pool of examples from section 2.2 to illustrate these concepts. In this chapter we introduce a rather sim ...
... Quotient spaces Up to this point in the text, when introducing new concepts for topological spaces such as Hausdorffness, compactness, connectedness etc, we have relied almost exclusively on the pool of examples from section 2.2 to illustrate these concepts. In this chapter we introduce a rather sim ...
Theorem of Van Kampen, covering spaces, examples and
... In particular, let us emphasize the following two cases. (i) If U ∩ V is simply connected, then π1 (X) � π1 (U ) ∗ π1 (V ); (ii) If V is simply connected, and N is the normal subgroup generated by the image (by ιU ∩V,U # ) of π1 (U ∩ V ) in π1 (U ), then π1 (X) � π1 (U )/N . Examples. (1) (Plane wit ...
... In particular, let us emphasize the following two cases. (i) If U ∩ V is simply connected, then π1 (X) � π1 (U ) ∗ π1 (V ); (ii) If V is simply connected, and N is the normal subgroup generated by the image (by ιU ∩V,U # ) of π1 (U ∩ V ) in π1 (U ), then π1 (X) � π1 (U )/N . Examples. (1) (Plane wit ...
Categories of certain minimal topological spaces
... Then A is both open and closed. Thus X—A is both open and closed. Since X—A is minimal Hausdorff, then X—A contains an isolated point which is also isolated in X. Thus A does not contain all the isolated points of X which contradicts our definition of A. COROLLARY 3. A compact space X where X is den ...
... Then A is both open and closed. Thus X—A is both open and closed. Since X—A is minimal Hausdorff, then X—A contains an isolated point which is also isolated in X. Thus A does not contain all the isolated points of X which contradicts our definition of A. COROLLARY 3. A compact space X where X is den ...
a hit-and-miss hyperspace topology on the space of fuzzy sets
... topology is also called as a hit-and-miss type of topology as it is generated by the sets which hit a finite number of open sets and misses a closed set. More precisely, if (X, d) is a metric space and CL(X) denotes the space of non-empty closed subsets of X then the Vietoris topology τV on CL(X) ha ...
... topology is also called as a hit-and-miss type of topology as it is generated by the sets which hit a finite number of open sets and misses a closed set. More precisely, if (X, d) is a metric space and CL(X) denotes the space of non-empty closed subsets of X then the Vietoris topology τV on CL(X) ha ...
Course 421: Algebraic Topology Section 1
... restriction of f to Ai is continuous for each i. But Ai is closed in X, and therefore a subset of Ai is relatively closed in Ai if and only if it is closed in X. Therefore f −1 (G) ∩ Ai is closed in X for i = 1, 2, . . . , k. Now f −1 (G) is the union of the sets f −1 (G) ∩ Ai for i = 1, 2, . . . , ...
... restriction of f to Ai is continuous for each i. But Ai is closed in X, and therefore a subset of Ai is relatively closed in Ai if and only if it is closed in X. Therefore f −1 (G) ∩ Ai is closed in X for i = 1, 2, . . . , k. Now f −1 (G) is the union of the sets f −1 (G) ∩ Ai for i = 1, 2, . . . , ...
Old Lecture Notes (use at your own risk)
... This unique topology τ is generated by the subbasis fi −1 (U ) : i ∈ I, U ∈ τi . Moreover, τ is the coarsest topology on X such that all the fi are continous. Proof. We first show uniqueness: suppose σ and τ are two topologies satisfying the Categorical Condition and consider the map idτ,τ : (X, τ ) ...
... This unique topology τ is generated by the subbasis fi −1 (U ) : i ∈ I, U ∈ τi . Moreover, τ is the coarsest topology on X such that all the fi are continous. Proof. We first show uniqueness: suppose σ and τ are two topologies satisfying the Categorical Condition and consider the map idτ,τ : (X, τ ) ...
Lecture notes
... space theory, that the closure is the set of limit points of a set. Unfortunately limits are not always defined in topological spaces, so this definition does not generalise directly. 5In fact, one can prove that any metric on Rn of the form d(x, y) = kx − yk where k·k is a norm (see http: //en.wiki ...
... space theory, that the closure is the set of limit points of a set. Unfortunately limits are not always defined in topological spaces, so this definition does not generalise directly. 5In fact, one can prove that any metric on Rn of the form d(x, y) = kx − yk where k·k is a norm (see http: //en.wiki ...
The Structural Relation between the Topological Manifold I
... The Structural Relation between the Topological Manifold Example 2.2 Let M be an open subset of Rn with the subspace topology then M is an n-manifold Example 2.3 The simplest examples of manifold not homeomorphic to open subsets of Euclidean space are the circle S 1 and 2-spheres S2 which way be de ...
... The Structural Relation between the Topological Manifold Example 2.2 Let M be an open subset of Rn with the subspace topology then M is an n-manifold Example 2.3 The simplest examples of manifold not homeomorphic to open subsets of Euclidean space are the circle S 1 and 2-spheres S2 which way be de ...
