X is infinite set with the finite complement topology. X is Hausdorff ??? why??? please thank...

X is infinite set with the finite complement topology.

X is Hausdorff ??? why???

please thank U

Expert Solution

No, X is not Hausdorff.

(A)We’ll prove that no two open sets are disjoint, which implies the result. BWOC, let
U, V be disjoint open sets. This implies

V ⊂ , and is a finite set, which implies V
is finite. But then is infinite, contradicting the fact that V is open.

(B) Let A ⊂ X and let
{U} be an open cover. Pick any single element of the cover, U∗ . It’s complement has finitely many
elements, so there are only finitely many elements of A, x1, . . . , xn, that are not in this set. For each
one, find a Ui containing xi. Then U∗ ,{Uαi}is a finite subcover.

(C) Any open set other than X

or ∅ is a set that is compact, but not closed.

Please rate my answer and comment for doubts.

orchestra answered 2 years ago

Let X be an uncountable set, let τf be the finite complement topology on X, and...

Let X be an uncountable set, let τf be the finite complement topology on X, and let τc be the countable complement topology; namely, we have τf ={U⊂X : X\U is finite}∪{∅}, τc={U⊂X : X\U is countable}∪{∅}, where “countable” means that the set is either finite or countably infinite (in bijection with the natural numbers). (a) What are the compact subspaces of (X, τf )? Are all compact subspaces closed in (X, τf )? (b) What are the compact subspaces...

Let X be the set of all subsets of R whose complement is a finite set...

Let X be the set of all subsets of R whose complement is a finite set in R: X = {O ⊂ R | R − O is finite} ∪ {∅} a) Show that T is a topological structure no R. b) Prove that (R, X) is connected. c) Prove that (R, X) is compact.

Prove that a subset of a countably infinite set is finite or countably infinite

Prove that a disjoint union of any finite set and any countably infinite set is countably...

Prove that a disjoint union of any finite set and any countably infinite set is countably infinite. Proof: Suppose A is any finite set, B is any countably infinite set, and A and B are disjoint. By definition of disjoint, A ∩ B = ∅ In case A = ∅, then A ∪ B = B, which is countably infinite by hypothesis. Now suppose A ≠ ∅. Then there is a positive integer m so that A has m elements...

Considering the set X = {1,2,3,4,5,6,7,8}, calculate a Tx topology on X such that the following...

Considering the set X = {1,2,3,4,5,6,7,8}, calculate a Tx topology on X such that the following two conditions are met: Tx has exactly 8 different elements, that is, | Tx | = | X | = 8 It is possible to find a subspace S of X with three elements, such that the relative topology Ts in S (with respect to Tx) has exactly 8 different elements, that is, | Ts | = | Tx | = | X |...

Please justify below questions (its a topology question) with valid arguments: 1. Show that an infinite...

Please justify below questions (its a topology question) with valid arguments: 1. Show that an infinite set is not finite. 2. Is there an infinite set which is not countably in finite? Thanks in advance.

Let X be a non-degenerate ordered set with the order topology. A non-degenerate set is a...

Let X be a non-degenerate ordered set with the order topology. A non-degenerate set is a set with more than one element. Show the following: (1) every open interval is open, (2) every closed interval is closed, (3) every open ray is open, and (4) every closed ray is closed. Please note: Its a topology question.

5. For each set below, say whether it is finite, countably infinite, or uncountable. Justify your...

5. For each set below, say whether it is finite, countably infinite, or uncountable. Justify your answer in each case, giving a brief reason rather than an actual proof. a. The points along the circumference of a unit circle. (Uncountable because across the unit circle because points are one-to-one correspondence to real numbers) so they are uncountable b. The carbon atoms in a single page of the textbook. ("Finite", since we are able to count the number of atoms in...

Cardinality State whether the following sets are finite, countable infinite or uncountable. Set of positive perfect...

Cardinality State whether the following sets are finite, countable infinite or uncountable. Set of positive perfect squares. Is it finite, countable infinite or uncountable? If it is countably infinite, set up the bijection between ℤ+. Negative numbers greater than or equal to -5. Is it finite, countable infinite or uncountable? If it is countably infinite, set up the bijection between ℤ+. Odd positive integers. Is it finite, countable infinite or uncountable? If it is countably infinite, set up the bijection...

Exercise 31: (General definition of a topology) Let X be a set and O ⊂ P(X),...

Exercise 31: (General definition of a topology) Let X be a set and O ⊂ P(X), where P(X) := {U ⊂ X}. O is a topology on X iff O satisfies (i) X∈O and ∅∈O; (ii) ?i∈I Ui ∈ O where Ui ∈ O for all i ∈ I and I is an arbitrary index set; (iii) ?i∈J Ui ∈ O where Ui ∈ O for all i ∈ J and J is a finite index set. In a general...

Question