Inductive reasoning
... three times higher than the greatest humpbackwhale spout (9 ft). Possible conjectures: The height of a blue whale’s spout is about three times greater than a humpback whale’s spout. The height of a blue-whale’s spout is greater than a humpback whale’s spout. ...
... three times higher than the greatest humpbackwhale spout (9 ft). Possible conjectures: The height of a blue whale’s spout is about three times greater than a humpback whale’s spout. The height of a blue-whale’s spout is greater than a humpback whale’s spout. ...
ON FAINTLY SEMIGENERALIZED α
... Definition 29. A graph G(f ) of a function f : (X, τ ) → (Y, σ) is said to be θ-sgα-closed if for each (x, y) ∈ (X × Y )\G(f ), there exist U ∈ sgαO(X, x) and V ∈ σθ containing y such that (U × V ) ∩ G(f ) = ∅. Lemma 30. A graph G(f ) of a function f : (X, τ ) → (Y, σ) is θ-sgαclosed in X × Y if and ...
... Definition 29. A graph G(f ) of a function f : (X, τ ) → (Y, σ) is said to be θ-sgα-closed if for each (x, y) ∈ (X × Y )\G(f ), there exist U ∈ sgαO(X, x) and V ∈ σθ containing y such that (U × V ) ∩ G(f ) = ∅. Lemma 30. A graph G(f ) of a function f : (X, τ ) → (Y, σ) is θ-sgαclosed in X × Y if and ...
the quotient topology - Math User Home Pages
... That is to say, a subset U ⊆ X/R is open if and only q −1 (U ) is open. (We leave it to the reader to prove that this is a topology.) Note that there’s something slightly different here. When we defined the subspace and product topologies, we proved that they were the coarsest topologies making cert ...
... That is to say, a subset U ⊆ X/R is open if and only q −1 (U ) is open. (We leave it to the reader to prove that this is a topology.) Note that there’s something slightly different here. When we defined the subspace and product topologies, we proved that they were the coarsest topologies making cert ...
3. Topological spaces.
... (3.9) Definition. Let X and Y be topological spaces. A map !!ψ : X → Y is called continuous if the inverse image ψ −1 (V ) of every open subset V of Y is open in X. The map is an isomorphism if there is a continuous homomorphism !!ω : Y → X which is inverse to ψ. That is ωψ = idX and ψω = idY . (3.1 ...
... (3.9) Definition. Let X and Y be topological spaces. A map !!ψ : X → Y is called continuous if the inverse image ψ −1 (V ) of every open subset V of Y is open in X. The map is an isomorphism if there is a continuous homomorphism !!ω : Y → X which is inverse to ψ. That is ωψ = idX and ψω = idY . (3.1 ...
Construction of Spaces
... Then, ( X , i * S ) is said to have the subspace topology. Prop. If W is a subspace of X and X is a subspace of Y , then W is a subspace of Y . Proof. Let j : W X and i : X Y be inclusions . (i j ) 1 (V ) j 1 (i 1 (V )) for all open sets V Y . All open sets in X have the form i 1 (V ), ...
... Then, ( X , i * S ) is said to have the subspace topology. Prop. If W is a subspace of X and X is a subspace of Y , then W is a subspace of Y . Proof. Let j : W X and i : X Y be inclusions . (i j ) 1 (V ) j 1 (i 1 (V )) for all open sets V Y . All open sets in X have the form i 1 (V ), ...
WHAT IS A TOPOLOGICAL STACK? 1. introduction Stacks were
... The significance of the above result is that the maps ωx are usually very easy to compute. As a simple application of the above theorem let us see in an example how we can determine if a given orbifold is a good orbifold (in the sense of Thurston). That is, whether it is uniformizable. Example 2.5. ...
... The significance of the above result is that the maps ωx are usually very easy to compute. As a simple application of the above theorem let us see in an example how we can determine if a given orbifold is a good orbifold (in the sense of Thurston). That is, whether it is uniformizable. Example 2.5. ...
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.