Mid-Term Exam - Stony Brook Mathematics
... Let x and y be two distinct points of X, and set D = d(x, y) > 0. Let f : X → R be the continuous function defined by f (z) = d(x, z). Since X is connected, f (X) is connected, too. Hence [0, D] ⊂ f (X). Thus f (X) is uncountable. Hence X must be uncountable, too. 2. Let f : X → Y be an injective co ...
... Let x and y be two distinct points of X, and set D = d(x, y) > 0. Let f : X → R be the continuous function defined by f (z) = d(x, z). Since X is connected, f (X) is connected, too. Hence [0, D] ⊂ f (X). Thus f (X) is uncountable. Hence X must be uncountable, too. 2. Let f : X → Y be an injective co ...
1 - Ohio State Computer Science and Engineering
... space X when the choice of T is clear. (In fact, we will mostly talk about the topology induced from a Euclidean space in this class.) Remark. The topology (as well as the induced topology) in Euclidean space is the most common topological space one will encounter. The definition of topological spac ...
... space X when the choice of T is clear. (In fact, we will mostly talk about the topology induced from a Euclidean space in this class.) Remark. The topology (as well as the induced topology) in Euclidean space is the most common topological space one will encounter. The definition of topological spac ...
(The Topology of Metric Spaces) (pdf form)
... If (S, d) is a metric space, we let T = TS be the set of open sets of the metric space. The set T is a collection of subsets of S that has the following properties: S (O1) If Ui ∈ T for i ∈ I, then i∈I Ui ∈ T ; (O2) If U, V ∈ T , then U ∩ V ∈ T ; (O3) ∅, S ∈ T . A collection T of subsets of a set S ...
... If (S, d) is a metric space, we let T = TS be the set of open sets of the metric space. The set T is a collection of subsets of S that has the following properties: S (O1) If Ui ∈ T for i ∈ I, then i∈I Ui ∈ T ; (O2) If U, V ∈ T , then U ∩ V ∈ T ; (O3) ∅, S ∈ T . A collection T of subsets of a set S ...
1. Basic Point Set Topology Consider Rn with its usual topology and
... I. Let X, Y be compact, connected, Hausdorff spaces. (A) Prove that a local homeomorphism f : X −→ Y is a covering projection. (B) Prove that an immersion h : Sn −→ Sn , where n ≥ 2 is a diffeomorphism. What happens if n = 1? II. Let A, X, Y be topological spaces. (A) If X is compact and Y is Hausdo ...
... I. Let X, Y be compact, connected, Hausdorff spaces. (A) Prove that a local homeomorphism f : X −→ Y is a covering projection. (B) Prove that an immersion h : Sn −→ Sn , where n ≥ 2 is a diffeomorphism. What happens if n = 1? II. Let A, X, Y be topological spaces. (A) If X is compact and Y is Hausdo ...
weak-* topology
... Let X be a locally convex topological vector space (over C or R), and let X ∗ be the set of continuous linear functionals on X (the continuous dual of X). If f ∈ X ∗ then let pf denote the seminorm pf (x) = |f (x)|, and let px (f ) denote the seminorm px (f ) = |f (x)|. Obviously any normed space is ...
... Let X be a locally convex topological vector space (over C or R), and let X ∗ be the set of continuous linear functionals on X (the continuous dual of X). If f ∈ X ∗ then let pf denote the seminorm pf (x) = |f (x)|, and let px (f ) denote the seminorm px (f ) = |f (x)|. Obviously any normed space is ...
On finite $ T_0 $
... and this topology is unique up to permutation. It was used only to obtain the following existence theorem. Theorem 1. 1f n is any finite positive integer, then there is a finite space X on n points and X has the fixed point property. Proof. In establishing this the topology used is the tower topolog ...
... and this topology is unique up to permutation. It was used only to obtain the following existence theorem. Theorem 1. 1f n is any finite positive integer, then there is a finite space X on n points and X has the fixed point property. Proof. In establishing this the topology used is the tower topolog ...
Final Answers
... space is homeomorphic to RP 2 #RP 2 . Using the same rule, abab ∼ a1 a1 b−1 b. But b−1 b can be glued together and eliminated, so the second symbol is equivalent to a1 a1 and the second space is homeomorphic to RP 2 . These two spaces are not homeomorphic. 5. Prove that [0, 1] is connected OR prove ...
... space is homeomorphic to RP 2 #RP 2 . Using the same rule, abab ∼ a1 a1 b−1 b. But b−1 b can be glued together and eliminated, so the second symbol is equivalent to a1 a1 and the second space is homeomorphic to RP 2 . These two spaces are not homeomorphic. 5. Prove that [0, 1] is connected OR prove ...
CATEGORIES AND COHOMOLOGY THEORIES
... Similarly, if products exist in ??‘?,one has a r-category SH@(S) associated to V with its product as composition: one defines e”(S) as the category of contravariant functors B(S) --f V which take disjoint unions to products. For a third example, if $9 is the category of modules over a commutative ri ...
... Similarly, if products exist in ??‘?,one has a r-category SH@(S) associated to V with its product as composition: one defines e”(S) as the category of contravariant functors B(S) --f V which take disjoint unions to products. For a third example, if $9 is the category of modules over a commutative ri ...
