(a) This exercise will give an example of a connected
space which is
not locally connected. In the plane R2
, let X0 = [0, 1] × {0},
Y0 = {0} × [0, 1], and for each n ∈ N, let Yn = {1/n} × [0,1]. Let
Y = X0 ∪ (S∞
n=0 Yn). as a subspace of R
2 with its usual topology.
Prove that Y is connected but not locally connected. (Note
that
this example also shows that a subspace of a locally
connected
space need not be locally connected even if it is closed.)
Hint: There are many ways to prove this. One of them is as follows:
Define for each n ∈ N, Zn = X0 ∪ Yn and let Z0 = X0 ∪ Y0. State why
Zi is connected for each i ∈ N ∪ {0}. State why Y =S∞
i=0 union Zi is connected. To show that Y is
not locally connected, examine the point (0,1/2).
(b) Prove that an open subspace of a locally connected space is
locally
connected.
(c) Consider Y = {0} ∪ {1/n: n ∈ N} as a subspace of (R , U). Use Y
to show that a continuous image of a locally connected space need
not be locally connected even if the function is a bijection.
Hint: The set N ∪ {0} is discrete in (R , U). A similar argument
used in part
(a) could be used to show that Y is not locally connected.
(a) This exercise will give an example of a connected space which is not locally connected. In th...
New problems for 2020 1. A topological space is called a T3.space if it is a T, space and for every pair («,F), where € X and F(carefull), there is a continuous function 9 :X (0,1 such that f(x) 0 and f =1 on F. Prove that such a space has the Hausdorff Separation Property. (Hint: One point subsets are closed.] 2. Let X be topological space, and assume that both V and W are subbases for the topology. Show...
want proof for theorem 7.12 using definition 7.9 Problem 7.7. Give an example of a space that is connected, but not path con- nected. Problem 7.8. Show that R" is not homeomorphic to R if n>1 Definition 7.9. Let be a point in X. Then X is called locally path connected at a if for each open set U containing r, there is a path connected open set V containing r such that V CU. If X is locally path...
Example: Let x, y ∈ Rn, where n ∈ N. The line segment joining x to y is the subset {(1 − t)x + ty : 0 ≤ t ≤ 1 } of R n . A subset A of Rn, where n ∈ N, is called convex if it contains the line segment joining any two of its points. It is easy to check that any convex set is path-connected. (a) Let f : X → Y be an...
2. For each space below, give an example of a set that does not span the indicated space. Explain why (a) The subspace (lc d) a b) The subspace | y | | x + y + z = 0 R 2
Consider R with the usual Euclidean topology and let I = [0, 1] be the closed unit interval of R with the subspace topology. Define an equivalence relation on R by r ~y if x, y E I and [x] = {x} if x € R – I, where [æ] denotes the equivalence class of x. Let R/I denote the quotient space of equivalence classes, with the quotient topology. Is R/I Hausdorff? Is so, prove so from the definition of...
Topology C O, 1 and be the supremum norm (a) Prove that (X || |) is a Banach space. You can assume that (X, | |) is a normed vector space (over R) |f|0supE0.1 \5(x)|.| 4. Let X C (b) Show that || |o0 that the parallelogram identity fails.] on X is not induced by any inner product. Hint: Check for all E[0, 1]. Show that {gn}n>1 (0, 1] BI= {gE X |9||<1} is a compact (c) For every 2...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
5. Let (Xn)n be a Markov chain on a state space S with n-step transition probabilities PTy = P(X,= y|Xo = x). Define (n) N x Xn=r n0 and U(G,) ΣΡ. n0 Show that (a) U(x, y)ENy|Xo= x] and (b) U(a, y) P(T, < +o0|X0= x)U(y, y), where Ty = inf {n 2 0 : X y}.
A topological space X has the Hausdorff property if cach pair of distinct points can be topologically scparated: If x, y E X and y, there exist two disjoint open sets U and U, with E U and y E U and UnU = Ø. (a) Show that each singleton set z} in a Hausdorff space is closed A function from N to a space X is a sequence n > xj in X. A sequence in a topological space...
Problem 4. Give an example of a linear operator T on a finite-dimensional vector space such that T is not nilpotent, but zero is the only eigenvalue of T. Characterize all such operators. Problem 5. Let A be an n × n matrix whose characteristic polynomial splits, γ be a cycle of generalized eigenvectors corresponding to an eigenvalue λ, and W be the subspace spanned by γ. Define γ′ to be the ordered set obtained from γ by reversing the...