a) Let X = { a, b, c,d, e }, if B = {{a,b},{d,c}, {e, f}}forms a basis for X, find the topology generated by B. b) Let B = {(a, b) : a,b e Q} be the collection of open intervals in R with rational endpoints. Show that: (1) B is a basis for some topology on R. (2) The topology generated by B is the usual topology on R.

a) Let X = { a, b, c,d, e }, if B = {{a,b},{d,c}, {e, f}}forms a basis for X, find the topology generated by B. b) Let B = {(a, b) : a,b e Q} be the collection of open intervals in R with rational endpoints. Show that: (1) B is a basis for some topology on R. (2) The topology generated by B is the usual topology on R.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

1. A topological space is said to be second countable if it has a countable basis. (a) Let R have the standard topology. Is RX R second countable? If yes, produce a countable basis. Otherwise, prove why such basis is not possible.
Let W = span{(2, 3, 0, – 1), (1,3,0, –1), (3,6,0, –2)}. a) Find a basis for W b) Find a basis for W+ c) Verify the dimension theorem for W and W+
- 1 1 -1 3) Let W be the space spanned by vectors -2 -1 0 -3 0 2 -2 -1 -1 -6 Then a basis for W would be:
5. Let X = R? and let В %3 { (а, b) х (с, d) с R? | a
SHow that the Collection (= { [91b) | a
6. Let A= | -2 -2 8 23-13 || ! | -| -4 a) Find a basis fowr Col A b) Find a basis tur Nul A math 3326 Page I c) Find dimcoLA) , dim (Nor A ) ,
Recall that R, denotes the Sorgenfrey line; the set of real numbers with the topology generated by the basis {[x, y): x < y}. Determine whether or not R, is connected. Justify your answer.
Q1:Let V be a finite dimensional vector space, I) if L ⊂ V be a linearly independent set in V. Suppose that ∣L∣ = dim(V). Prove that L forms a basis for V. II) if S ⊂ V be a finite spanning set for V. (i.e. S is finite, and span(S) = V.) Suppose that ∣S∣ = dim(V). Prove that S forms a basis for V.
Wich of the following statements are true? Select ain true statements! Only select true statements! (a) Any linearly independent set that has the same number of elements as the dimension of the space is a basis. (b) The set {(1.0, 1). (0, 1, 0). (1, 1, 1). (4, 4, 4)} spans R³. (c) P, has a linearly independent subset with 5 elements. (d) Any subset that spans a vector space V must have at least dim(V) many elements. (e) For v; e V if £-, \;V; E- Hiv; then A; = li for all i = 1, ..., n. Li=1 Li=1 (a) C (b) a (c) d)
C) Let (X. t) be a topological space and A CX. If S is a sub-basis for the topology T, then S = (AnG;GES) is a sub-basis for the relative topology TA-
Prove that R with the standard topology has a countable basis, and R × R with the standard topology has a countable basis.
If W is the sub space can find and orthogonal basis for a W