Topology Boundry Question

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• Mar 1st 2009, 03:47 PM
flaming
Topology Boundry Question
For each $n\in Z_+$, let $B_n = {n, n+1, n+2, ....}$ and consider the collection B = ${B_n | n\in Z_+}$

(a) Show that B is a basis for a topology for $Z^+$

(b) Show thte the topology on X generated by B is not Hausdorff

(c) Show that the sequence (2,4,6,8,...) converges to every point in $Z_+$ with the topology generated by B
• Mar 1st 2009, 06:56 PM
aliceinwonderland
Quote:

Originally Posted by flaming
For each $n\in Z_+$, let $B_n = {n, n+1, n+2, ....}$ and consider the collection B = ${B_n | n\in Z_+}$

(a) Show that B is a basis for a topology for $Z^+$

(b) Show thte the topology on X generated by B is not Hausdorff

(c) Show that the sequence (2,4,6,8,...) converges to every point in $Z_+$ with the topology generated by B

(a) The followings are the basis elements of B.

B1={1,2,3,.................}
B2={2,3,4,...............}
B3={3,4,5,............}
........
Bn={n, n+1, n+2,,,,}
...........

To check B is a basis for Z+, you first need to check the union of the above elements are indeed Z+.
Second, you need to verify that for each x in the interesection of the basis elements, there exists a basis element containing x and contained in the intersection.

For instance, consider the intersection of B1 and B2. For each x in the section, you can find a basis element that containing x.

(b) Pick two numbers, for instance, 3 and 5. Check open sets containing 3 & 5 respectively. Those open sets cannot be chosen disjointly.

(c) Every open set containing a number in Z+ contains all but finite numbers of the above sequence in your topology. Thus, it converges everywhere in Z+.