do any of them please? Determine the accumulation points of the following sets o
ID: 2987616 • Letter: D
Question
do any of them please?
Determine the accumulation points of the following sets open interval(a,b), a,b eR* Infinite open interval (a,infinity),a e R R Determine those subsets of R* Which have p as a limit point Let T1 and T2 be topologies on a set X with T1 coarser than T2, ie Show that every T accumulation point of a subset A of X is also a T1-accumulation point Construct an example in which the converse of (i) does, not hold. CLOSED SETS, CLOSURE OF A SET, DENSE SUBSETS Construct a non-discrete topological space in which the closed sets are identical to the open sets. Prove: Construct an example in which equality does not hold. Prove: . Construct an example in which equality does not hold. Prove: If A is open, then Prove: Let A be a dense subset of (X.T). and let B be a non-empty open subet of X. Then Let T1 and T2 be the topologies on X with T1 coarser than T2, Show that the T2-clousre of any subset A of X is contained in the T1-clousre of A Show that every non-finite subset of an infinite cofinite space X is dense in X. Show that every non-empty open subset of an indiscrete space X is dense in X INTERIOR,EXTERIOR,BOUNDARY Let X be the descrete space let find int(A) ext(A) and b(A) Prove if and only if A is closedExplanation / Answer
A simple way is to do a calculational proof, starting at the most complex side, to calculate which elements are in (A?B)?(A?C): for all x,
?(?)??x?(A?B)?(A?C)"definition of ?; definition of ?, twice"(x?A?x?B)?(x?A?x?C)"logic: simplify by 'factoring out' x?A, using the fact that ? distributes over ?"x?A?(x?B?x?C)"definition of ?; definition of ?"x?A?(B?C)
Now by extensionality (i.e., equal sets have the same elements) it follows that
(0)A?(B?C)=(A?B)?(A?C)
Note how we didn't need to prove two separate ? cases. Also note that the key step (?) uses the logical law
(1)P?(Q?R)?(P?Q)?(P?R)
We see that (0) and (1) have the same structure. Since (1) is in the logic domain, it is more generally useful than (0), which is only applicable when dealing with sets.
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