Thread: Proving compactness and connectedness for this set

Could someone please help me with how to do this problem?

Given:
S1 = {(x,0):0<x<=1}
S2 = {(1/n,y): n=1,2,3,.... and 0<=y<=1}
S3 = {(0,1/2)}
Consider the sets:
S = S1 in union with S2
S0= S1 in union with S2 in union with S3

1) Use the Bolzano-Weierstrass Theorem to prove that S0 is not compact.
2) Is the set S arcwise connected? (Justify)
3) Is the set S0 connected? (Justify)

Thanks a lot.

Originally Posted by AKTilted

Could someone please help me with how to do this problem?

Given:
S1 = {(x,0):0<x<=1}
S2 = {(1/n,y): n=1,2,3,.... and 0<=y<=1}
S3 = {(0,1/2)}
Consider the sets:
S = S1 in union with S2
S0= S1 in union with S2 in union with S3

1) Use the Bolzano-Weierstrass Theorem to prove that S0 is not compact.
2) Is the set S arcwise connected? (Justify)
3) Is the set S0 connected? (Justify)

Thanks a lot.

Slight hints and then you show some work:

Draw the set S_0> can you see a rather obvious set of points in converging to a point NOT in S0 (look closely at S2)? this is (1)
If you get a good grip of the set S0 then you msut have a "feeling" what point there can't be joined with other points there by an arc (and this is (2))
S0 is connected: assume S0 = A \/ B with A,B open . Some of these sets must contain the point in S3. prove this set is all of S0 and you're done.

Tonio

Originally Posted by tonio

Slight hints and then you show some work:

Draw the set S_0> can you see a rather obvious set of points in converging to a point NOT in S0 (look closely at S2)? this is (1)
If you get a good grip of the set S0 then you msut have a "feeling" what point there can't be joined with other points there by an arc (and this is (2))
S0 is connected: assume S0 = A \/ B with A,B open . Some of these sets must contain the point in S3. prove this set is all of S0 and you're done.

Tonio

...........ah...I think S is arcwise connected, and So is not connected...


I mean..for 3), S1 and S2 is connected..and S2 and S3 is connected..but..S1 and S3 is not connected..would the whole thing still be connected??

Originally Posted by wxyj

...........ah...I think S is arcwise connected, and So is not connected...


I mean..for 3), S1 and S2 is connected..and S2 and S3 is connected..but..S1 and S3 is not connected..would the whole thing still be connected??

Not necessarily since their intersection is empty.

Tonio