Is not closed set in X x X. Then the space X is: (a) Hausdorff Space (b) Not Hausdorff Space (c) T1-space (e) Not T1-space
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- let (X,T) be a topological space. Then a function f is continuous at x0 element of X if and only if f is both lower semi continuous and upper semi continuous at x0 element of X.Let (X, τ1) be any topological space and (X, τf ) be the finite complement topological space.Show that the space (X, τ1) is a T1−space if and only if τf ⊆ τ1.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
- Define what it means for a topological space to be completely regular,Let (X, τ) be the topological space and A⊂X. In this case, show it as in the picture.Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. Conclude that, with this metric, a subset of ℝ3 is sequentially compact if and only if it is closed and bounded.
- Prove or disprove that (R, Tco-finite) is T2- spaceProve that an affine map is continuous if and only if its linear part is continuousProve the Theorem 5.5.7 - Let (X1, d1) and (X2, d2) be metric spaces and let f : X1 → X2. Then f is continuous on X1 iff f-1 (G) is an open set in X1 whenever G is an open set in X2.
- Let (X, T) and (Y, T1) be two topological spaces and let f be a continuous mapping of X into Y. If (Y, T1) is a T1 space, then (X, T) is a T1 space?Prove that if f is a continuous mapping of a compact metric space, X, into a metric space Y, then f(x) is compact.True or False: a)Every subset of a topological space is either open or closed.b)If X is a topological space with the discrete topology and if Xhas least two elements, then X is not connected.c) True or False: If X is a topological space, then there always is a metric on Xwhich gives rise to its topology.d) True or False: If X and Y are topological spaces and if f : X → Y is a constantmap (which means that there is a point y ∈ Y such that f(x) = y for all x ∈ X),then f is continuous.e) True or False: If X is a topological space, then X is both open and closed