Exercise 5 Let X = (a, b, c, d, e) and let T = {X, 0, (a), (c,d), {a,c,d), (b, c, d, e]) is a topology on X. Find the closure A, the interior Å, the exterior Ext(A) and the boundary b(A) of the following sets: A₁ = 0, A₂ = X₁ A₂ = {a, c) Soluti
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- Let M be a closed subset in (X,d). Show that b(M) is a subset of b(b(M)) where b(M) is the boundary of M.2. Let (Vr) be a topological space. Where Y ={a,6 ,¢ ,d,e } andr={X .®,{c},{d}. {ed} .{d.e} .{e.d.e}, {b,c,a}, {a,b,c,d }}Show that f° ={ {c.d},{d,e},{a,b.c}} is a subbase for thetopology +r. 3. Let X ={a,b,c,d,e}, f° ={ {a,b} , {b,c} ,{c,e},fe} } o PX)Find the topology + on X generated by /".Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed subset of ℝ2
- Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?I, Let ¥ ={a,b,c} and B={ {a,c} ,{b,.c} } c P(X). Show thatcannot be a base for any topology r on X . 2. Let (Vr) be a topological space. Where Y ={a,6 ,¢ ,d,e } andr={X .®,{c},{d}. {ed} .{d.e} .{e.d.e}, {b,c,a}, {a,b,c,d }}Show that f° ={ {c.d},{d,e},{a,b.c}} is a subbase for thetopology +r. 3. Let X ={a,b,c,d,e}, f° ={ {a,b} , {b,c} ,{c,e},fe} } o PX)Find the topology + on X generated by /".Let X = {a, b, c, d} with the following topology: ?= {X,0, {a},{b}, {a,b}} Let Y={x,y,z} Consider the following quotient map: p: X→Y with p(a)=x p(b)=x p(c)=y p(d)=z. What are the open sets in Y with the quotient topology?
- Let X={a,b,c,d} Let T1={X, emptyset, {a}, {a,c}, {b}, {b,c}, {a,b},{a,b,c}, {b,c,d}} T2={X, emptyset, {b}, {b,d}, {c}, {c,d\}, {b,c},{b,c,d}, {a,b,c}} how that the T1 and T2 topologies are topologically equivalent (you can give the homeomorphism or show that they are by their bases)Prove that if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)} is bounded above by (1/3).Consider the discrete topology τ on X:={a,b,c,d,e}. Find subbasis for τ which does not contain any singleton sets.