Show that is not compact for the Standard topology but is compact for the cofinite topology.

Q: Let! aset os R, sho that whe ther Tbe acofinite to pology on (R,T) is To-spuce and TisPuceand or not…

A: The given question is related with topology. Given that Τ is the co-finite topology on R . We have…

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: Find a base for the Standord tapology-

A: This is a question of topology.

Q: 2) Show that the digital line topology on Z is not Hausdorff.

A: Given that Digital line topology on ℤ, we have to prove that it not Hausdorff Therefore, Let a space…

Q: Find two examples for equivalet norms on three dimensional real space R³. .

A: To take two vectors whose norms are the same.

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: Suppose (X, d) is a nonempty metric space with a discrete topology. Show that X is connected if and…

A:

Q: show that (0.1)with usual topology is connected

A: Sol

Q: 6. Let f: R? → R? be a continuous function. Prove that f ([0,1] × [0,1]) is compact.

A:

Q: Give a basis for the usual topology for R. Justify your answer

A: Given that R(set of all real numbers) with usual topology.

Q: 3) Let R be the set of all real numbers with the euclidean topology and f: R → R be continuous…

A: Yes, the image of a continuous function from a path-connected space is always path-connected.…

Q: Prove that the set of mxn upper triangular matrices is a subspace of Mmxn(F).

A: We have to prove that -

Q: True or False: Consider the subsets (0, 1) and (2, 3) of R, the set of all real numbers with the…

A: The given statement is  Consider the subsets (0, 1) and (2, 3) of R, the set of all realnumbers with…

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

A:

Q: Provide an example of a topological space that is connected but not path-connected,why?

A: We use definition of connected and path connected sets. We use contradiction method.

Q: Use only the definition of affine dependence to show that an indexed set {v1, v} in R" is affinely…

A:

Q: if

A: R is not connected if T is the finite closed topology.

Q: Show that if a topological space has a finite number of points each of which is closed then it has…

A: Topology question.

Q: Prove that Cl(Q) = R in the standard topology on R. %3D

A: The closure of S may equivalently be defined as the union of S and its boundary, and also as the…

Q: Find the boise for the in- topology. Find a base for the Standord tepology-

A: This is a question of topology.

Q: A topological space (X, t) is a T1-space iff t contains the cofinite topology on X.

A:

Q: Give an example of a norm space.

A:

Q: Show that topologically completeness is weak hereditary property.

A: In this question, we have to show that completeness is a weak hereditary property.

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

A:

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: 0 0 0 0

A:

Q: Show That R is Banach Sface.

A: To Show -

Q: Let X = R, and let T consist of 0, R, and all intervals of the form (a, b) where a, b e R and a < b.…

A: A topological space is an ordered pair {\displaystyle (X,\tau ),} where {\displaystyle X} is a set…

Q: 2. Let (X, 1) be any topological space and (X, Tf) be the finite complement topological space. Show…

A: Let (X, τ1) be any topological space and (X, τf ) be the finite complement topological space.Show…

Q: e Guen an example to shus thet the inege lincarly independant set under a Lucar may mapping not be…

A: We have to give an example of a linear mapping which maps linearly independent set to linearly…

Q: Suppose X has the discrete topology. Then the infinite product X with the product topology is also…

A: It is given that X has the discrete topology. Check whether the infinite product Xw with product…