Let T be a topology on X. Assume that T is Hausdorff and let x € X.(1) Show that {x} (and hence every finite set) is always closed.(2) Show, by an example, that the separation hypothesis cannot be dispensed with.(3) Give an example of a non-separated space in which singletons are closed.(4) Show that the intersection of all open sets containing r is {x}.(5) Give an example of a non-separated space in which the previous result holds.

let T be a topology on x. T is a Hausdorff space

Answered: Let T be a topology on X. Assume that T…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 21E: Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by...

See similar textbooks

Similar questions

Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
Give an example of a set X and topologies T1 and T2 on X such that T1 union T2 is not a topology on X
For any infinite set X, the co-countable topology on X is defined to consist of all U in X so that either X\U is countable or U=0. Show that the co-countable topology satisfies the criteria for being a topology.
Let T and T 'be two topologies of a set X.Is the family T U T´formed by the openings common to both also a topology of X?
let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B belongs to t with x= A union B and A intersect B = zero
Prove that T is the discrete topology for X iff every point in X is an open set
Let (X,T) be a topological space Property C=P.C. A subset A of x has P.C If it's subset of the union of two disjoint nonempty open subsets of X then A is contained in only one of these open sets. Prove the following; If A and B have P.C and A̅̅∩B≠ø then A∪B has property c.
If X is a metric space with induced topology Ƭ, then (X,Ƭ) is Hausdorff. The contrapositive of this theorem must be true:If (X,Ƭ) is not Hausdorff, then X is not a metric space. 1) Consider (ℝ,Ƭ) with the topology induced by the taxicab metric. Using the definition for Hausdorff, give an example of why (ℝ,Ƭ) is Hausdorff. 2) The finite complement topology on ℝ is not Hausdorff. Explain why ℝ with the finite complement topology is non-metrizable.
Consider the following topological spaces: Space X: A closed unit interval [0, 1] with the usual Euclidean topology. Space Y: A set of real numbers R with the co-finite topology, where a subset U of R is open if and only if its complement in R is finite or equal to R. Question: Determine whether the topological spaces X and Y are homeomorphic. Justify the answer.
Let X be a set equipped with the finite complement topology show X is compact Show that X is compact Hausdorff if and only if it has the discrete topology.
Let A be a closed bounded subset of a metric space (X, d). Is A acompact subset? Explain why.
Prove that ℝ with lower limit topology is Hausdorff

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

Let T be a topology on X. Assume that T is Hausdorff and let x € X. (1) Show that {x} (and hence every finite set) is always closed.