Let T_us be the usual topology on R and T_|| be the lower limit topology generated by the unions of {]a,b]/ a,bER;asb}. Define f the mapping from (R, T_us) into (R, T_II) by f(x) = 1 and g the mapping from (R, T_II) into (R, T_us) by g(x) = 4x. Then * f and g are both homeomorphisms None of the choices g is a homeomorphism but f is not f is a homeomorphism but g is not

Q: Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such…

A: First, we will find the equation of a line given two points.

Q: Let R = - b ,be Z}and let ø : R→Z be a mapping defined by : b = a-b. a a

A: We will find out the required value.

Q: Prove that ATM <m EQtm: (a) Construct a mapping reduction from ATM to EQtm. (b) Prove that the…

A:

Q: Theorem: If R is a fuzzy relation on the set X, then (R-1)-1 = R

A:

Q: Let f: X->Y and g:Y->X be maps between sets. Suppose that g ∘ f is equal to the identity map id_X:…

A: Given f:X→Y and g:Y→X be the maps between the sets.  Suppose that g∘f=idx. Then We need to prove or…

Q: Show that the Lie bracket defined on L/I is bilinear and satisfies the axioms (L1) and (L2) (Algebra…

A: To show that the Lie bracket defined on is bilinear and satisfies the axioms (L1) and (L2) of the…

Q: Let (X,m, µ) be a major space and (Y,?) be a topological space. Let f:X→Y be a function. Assume…

A: Let (X,m, µ) be a major space and (Y,?) be a topological space. Let f:X→Y be a function. Assume…

Q: Let X be a locally compact Hausdorf f space

A:

Q: Q3/(a) Let d be a trivial metric on a set of X, find the induced topology on X by a metric d. (b)…

A:

Q: The relation R is defined on R^2 by (x1, y1) ∼ (x2, y2) if and only if x1 − x2 ∈ Z,for (x1, y1) and…

A: a. The relation R is defined on R^2 by (x1, y1) ~ (x2, y2) if and only if x1 − x2 ∈ Z,for (x1, y1)…

Q: 4) Let T be the Euclidean topology on R and T1 be the discrete topology on R. Define f : (R, 71) →…

A:

Q: Exercise 3. In this exercise (X, d) is a metric space. .} 1) Assume that X is compact. For every r>…

A:

Q: Give an example of a closed subset of R that is not sequentially compact -Give an example of a…

A:

Q: Let T us be the usual topology on R and Tl be the lower limit topology generated by the unions of…

A:

Q: Let ƒ: R² → R² be an isomorphism where ([-])-[²] and f([:]) = [] = Find f ¹ƒ( [7¹]) X 0 2 2 0

A:

Q: The linear mapping A:R^2[x] -&gt; R^2x2 is defined by the rule Ap= [p(0);p(1);p(-1);p(0)]. Then the…

A: Step 1:A:R2[X]→R2∗2  Defined by  A(p(x))=[p(0)p(−1)​p(1)p(0)​] Step 2: let kernel K is the kernel of…

Q: Q. Let X = {1,2, 3, 4},T = {ø, X, {4},{1,2}, {1,2, 4}} and Y = {r, s, p, q}, F = {ó,Y, {p}, {r, s},…

A:

Q: let ( X Td)be Toplog9 me trics spaces the n EVery singteon set is closed set

A: Given that X,Td is a Topology Metric spaces.

Q: why is f~ homotopic? the question is asking for f.

A: Please find the answer in next step

Q: Find oxthogonal uncdary diagonalizing an or matri'y to lMowing tor One 2 (a) (b) 3 -2 -2

A: A square matrix A is called as orthogonal matrix if AxAT=ATxA=I where AT transpose of matrix A. and…

Q: Exercise 8. Show that a function F: R" → R" preserves the norm || · || if and only if it preserves…

A: We have the function F:ℝn→ℝn that preserves the norm · This implies that this function satisfies all…

Q: Q₂/(a) Let X=(1, 4, 5,7, 8, 9) and o= {(1,4),(4,7, 8}}, show that whether o is: (1) a subbase for a…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: 12 Let (RTR) is topological Space Such that TR-243 U2GSR=R-G-G is finste} Find buz), Z², Z. z' then…

A: Here given ℝ, Tℝ is a topological space such that Tℝ=ϕ∪G⊂ℝ: ℝ-G=Gc is finite This topology is known…

Q: Let T_us be the usual topology on R and T_I| be the lower limit topology generated by the unions of…

A: A homeomorphism is a bijective map whose inverse is also continuous

Q: Let: f : R → R with f(x) = 2x ^2 + 1. Prove or disprove that f is injective (one-to-one).

A:

Q: possible

A:

Q: metoix space and d[211?(₂) for all ₁2 ER 2,12₂ 17. (*/ 1²(₂) d is matrie X. let (Xid) be d' XXR…

A:

Q: Lot X= [a,b] with the standard topology and let Y= [9₁6) with the Cofinito topology. Why is the…

A:

Q: Let T_us be the usual topology on R and T Il be the lower limit topology generated by the unions of…

A: Homeomorphism is a bijective continuous map whose inverse is also continuous

Q: Exercise 1. Give examples of subsets A, B in a topological space X (for example X = R or X = R²)…

A: Disconnected sets

Q: Let T us be the usual topology on R and T_Il be the lower limit topology generated by the unions of…

A:

Q: Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f:…

A: We have to give an example of a topological space X such that the diagonal is not closed in X×X