Show that the image of an open set under a continuous mapping need not be open

Q: If  X is a countable set show that every topology on X is second countable.

A: To Show: If  X is a countable set show that every topology on X is second countable.

Q: show that the Cantor set is a closed subset of R and has measure zero.

A: We know the definition of Cantor set, The Cantor set is defined as the intersection C = ∩n=1∞Cn ,…

Q: show that any infinite discrete topological space is not compact

A:

Q: A topological space is Hausdorff iff limits of all nets in it are unique-

A: This is a problem of Topology.

Q: a) Show that every closed subset of a compact space is compact but the converse need not to be true…

A: As per the guidelines I had given the proof of the first theorem. Kindly repost the another theorem…

Q: be a closed subset of space Xa

A:

Q: Prove it. Establish the mapping 7 and IN.

A:

Q: Prove a neighborhood is an open set

A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: Exercise 2.46. Show that: (1) The topology generated by a subbasis is a topology. (2) The sets in…

A: 2.46

Q: Provide a direct proof that a sequentially compact set must be both closed and bounded without…

A: WE GIVE A DIRECT PROOF. First, we prove that set A is closed then A is bounded.

Q: that the boundary of a closed set is nowhere dense in

A: Matric space

Q: (b) Show that the continuous image of compact set is compact but the convers need not to be true.…

A:

Q: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

A:

Q: Show

A: We prove this statement by using definition of Borel Set.

Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U ·. U Bn equal A exactly?…

A:

Q: Prove that an affine map is continuous if and only if its linear part is continuous

A: If T:A→Rk is the affine map, where A is the affine span of an affine independent subset of Rn. Let…

Q: Use the Heine-Borel theorem to give an example of a set not compact in R with the usual topology…

A: An example of a set not compact in R with the usual topology and a non-compact set R2 with the usual…

Q: Show that any finite union of Nowhere Dense Sets is Nowhere Dense.

A:

Q: prove that Cantor set is closed and compact.

A: Use the property that arbitrary intersection of closed set is closed and the Heine Borel Theorem.

Q: A relation ''<'' is 1) İrreflexive. 2) Transitive. called a quasi-order on a set S if its: صواب ihi

A: We know that

Q: Please define compact space and the clousure of .a set

A:

Q: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

A: Given the usual topology on ℝ3. i.e. τusual=U⊆ℝ3: ∀x∈U, ∃∈&gt;0 such that B∈x⊆U

Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U... U Bn equal A exactly?…

A:

Q: Show that any convex open set is simply connected.

A: The objective is to prove that any convex open set is simply connected. Simply connected is nothing…

Q: Exercise 2.27. Show that a set U is open in a topological space X if and only if every point of U is…

A: Let, U is an open set in a topological space X. Then, U⊂U  So, U⊂int(U) and int(U)⊂U Hence, U=int(U)

Q: Find a topology in which the point x = 5 is an open set.

A:

Q: Use the definition of compact set to prove that the set [0,1] is compact

A: We know that a set E is compact if every open cover of E has a finite subcover. We suppose, that O…

Q: Show that the lower limit topology on R is strictly finer than the standard topology on R. That is:…

A:

Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: A finite set in a metric space cannot have accumulation points.

A: We have to prove that A finite set in a metric space can not have accumulation (Limit) point.

Q: Prove that the image of a continuous mapping of a compact set is compact.

A:

Q: What is the minimum number of points that can be contained in a space

A: There is no minimum number of points to be contained in a space. But if we want to find a location…

Q: Prove that derived set of a bounded set is also bounded.

A:

Q: Prove that a nonempty set is bounded if and only if it is contained in a bounded interval

A: A non empty set S is called bounded if it has both upper and lower bounds, In a certain sense a non…

Q: Determine the dimension of the space spanned by the set

A: Given,               S=12-13, 1002, 28-48, 1111, 3306

Q: Prove that the open set (0,2) is not a compact subset of ℝ by finding an open cover with no finite…

A: To prove:  (0, 2) is not a compact set of ℝ.

Q: Show that any subset of bounded set is bounded

A:

Q: The closare of a set is smallest closed set containing s صواب ihi

A: Let S be a subset of X: S⊆X. Let M be the closure of S. Then M is the smallest closed set which…

Q: Let ||.|| and ||..|| be two norms defined on space X.Show that ||.|| and ||..|| are equivalent if…

A:

Q: open

A:

Q: The intersection of any countable family of open dense-sets in a locally compact space X is dense.

A:

Q: Find an example of on a set X (G has or H of G which has le

A: Given: that Group G is active transitively on set X . G has only one orbit and subgroup H of G has…

Q: The closure of a compact subject of a regular space is compact.

A:

Q: Every proper simply connected oper equivalent to an open subset of the

A:

Q: Show that X is connected if and only if the only clopen subsets are Ø and X.

A:

Q: Show that every closed subset F of R' is the intersection of a countable col- lection of open sets.

A: detailed explanation mentioned below.

Q: Show that the closure of a connected space is connected.

A: