(b) Show that the continuous image of compact set is compact but the convers need not to be true. (c) Show that a space X is compact iff every family of closed subsets of X with finite intersection property has non-empty intersection. (4M)

Q: (c) Let X, Y and Z be sets. Prove that (X\Y)n (X \ Z) = X \ (YUZ).

A: We prove the given statement by using set theory.

Q: Show that the space (P is complete.

A: The given problem is to show that the space lp is complete. we can use properties of lp spaces to…

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

A:

Q: Elaborate on the role of duality in topological spaces.

A: Introduction:Topology is a branch of mathematics that studies the properties of spaces and the…

Q: Show that compact sets are closed under arbitrary intersections and finite unions. (Here we're…

A: To show that compact sets are closed under arbitrary intersections and finite unions

Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

A:

Q: Let X be a topological space. Prove that (a) Ø and X are closed. (b) Intersections of any number of…

A: Union of any open sets is open and intersection of two open sets is open.

Q: Q6) Find a topological space and a compact subset whose closure is not compact. Note hat a space is…

A:

Q: Prove that B = (0,1) is NOT a compact set.

A: To prove that (0,1) is not a compact set.

Q: Prove that (2,5) is an open set

A:

Q: Explain with an example of a topological space which is singleton that is open but not close and…

A: Example in refer of not  open but closed .Take , Here A is singletone set which is closed but not…

Q: prove or dis Prove a Every convex set of avectory is affine set- • Every affine set is sub space

A: A convex set is a subset of a vector space (such as Euclidean space) where for any two points within…

Q: The intersection of any countable family of

A:

Q: Define the Cartesian product of sets X and Y? How is this Cartesian product denoted?

A:

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

A:

Q: >) Recall that the symmetric difference of two sets X and Y, denoted XAY = (X-Y)U(Y-X), is the set…

A:

Q: Q1) Prove or Disprove: Let L(F) be a linear space and A,B are two convex subsets of L. Then, AXB is…

A: The problem is related with convex set. Given that,                 LF  is a linear space  and A, B…

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: Iwant to define these Poin (Defined) 1-) sobolve spaces 2-) Reflexive real Banach SPace

A:

Q: Prove the empty set hasn't supremum and infimum

A: Lower Bound: A lower bound of a subset A of a partially ordered set (X,⩽) is an element a∈X such…

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

A:

Q: If A and B are distinct points in 3-space then |AB| = - Select one: True False

A:

Q: Quertion:- Show that R is a Banach Space.

A:

Q: Define what it means for a topological space to be completely regular,

A:

Q: Show that the interval [0,1] is a closed set in R

A:

Q: Finite topological spaces are always complete. True False

A: Finite topological spaces are always complete. So it is true statement.

Q: ACX A + Ø and A + X. Let B = AU fr(A). .show that for the topology T = {X,Ø, A, B} it happens that…

A: Given A⊂X where X is non-empty and A≠X Suppose, T=X,∅,A,B be a topology.  Here B=A∪frA To prove for…

Q: Let R, S, and T be sets. Let f: R -» S, and g: S -» T be maps. Assume we know that qf is 1-1 Must f…

A: Given:

Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

Q: Find isomorphic spaces O R and R² O P3 and M2x1 O P3 and M2×3 O P3 and R2 O R' and M2×2

A: We have to choose whether the given spaces are isomorphic or not.

Q: Q1: (a) If (M, d) be a metric space and A M. Prove that A is the largest open set in M that…

A: The interior, or (open) kernel, of A is the set of all interior points of A: the union of all open…

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

A:

Q: ive example to show that a quotient space of a second countable space need not be second countabl

A: This is a problem of topology.