Let X,Y be topological spaces, and f:X->Y a homeomorphism, I.e. f is one-to-one, onto, and f and f-1 are continuous.
a.) Prove that if X is T4, so is Y.
b.) Prove that if X is separable, then so is Y.
Let X,Y be topological spaces, and f:X->Y a homeomorphism, I.e. f is one-to-one, onto, and f...
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous, Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
We used definition of homeomorphic as follows. If X and Y are topological spaces, a function f: X to Y is called homeomorphism if 1. f is continuous 2. f is bijective 3. inverse of f is continuous And in this case, we say that X is homeomorphic with Y. Thank you ! infinite) (5) Prove that all semiopen intervals in R (finite or homeomorphic are infinite) (5) Prove that all semiopen intervals in R (finite or homeomorphic are
l maps is a quotient map. 4, Let ( X,T ) be a topological space, let Y be a nonempty set, let f be a function that maps X onto Y, let U be the quotient topology on induced by f, and let (Z, V) be a topological space. Prove that a function g:Y Z is continuous if and only if go f XZ is continuous. l maps is a quotient map. 4, Let ( X,T ) be a topological...
1- Prove or disprove. (X,Y are topological spaces, A, B are subsets of a topological space X, Ā denotes the closure of the set A, A' denotes the set of limit points of the set A, A° denotes the interior of the set A, A denotes the boundary of the set A.) (a) (AUB) = A'U Bº. (b) f-1(C') = (F-1(C))' for any continuous function f :X + Y and for all C CY. (c) If A° ), then A°=Ā.
2. Let (X, dx), (Y, dy) be two metric spaces, and f:X + Y a map. (a) Define what it means for the map f to be continuous at a point x E X. (b) Suppose W X is compact. Prove that then f(W) CY is compact.
a) Prove the following theorem: Let f:(x,d)-(Y,p) be bijective and continuous. Then f is a topological mapping iff: VUCX: U open = f(U) open in Y. b) Þrove the following theorem: Let f :(X,,d) (X ,d) and f:(X2,d)) (X 3,d) be topological mappings, Then f of, (the composition of the two functions) is topological.
Suppose that f : X → Y is a continuous and surjective map between two topological spaces. Determine if the following statements are true or false. If true, prove the statement, if false, give a counter-example. (a) If X is path-connected, then so is Y. (b) If X is locally compact, then so is Y. (c) If X is Hausdorff, then so is Y.
26. Group D: If f : A + B is a function between metric (or topological) spaces such that f-1(U) is closed in A whenever U is an open set in B. Is f injective (that is, one-to-one)? Continuous? A homeomorphism?
Let S = {(x, y) = RP.22 + y2 = 1} denote the unit circle in R2 with the subspace topology. Define the function F: (0,1) + S via th (cos(2), sin(24t)) Prove that F is one-to-one, onto, and continuous, but not a homeomorphism.
3. Let X and Y be two topological spaces, and let AC X andBCY. T'hen there are two topologies on Ax B: (a) the subspace topology on A × B C X x Y, where X × Y is equipped with the product topology; (b) and the product topology on A x B, where A S X and BSY are equipped with the subspace topologies. Show that these two topologies are equal