Show that given subsets are not compact by describing....

[luckyc1423]

1) Show that each subset of R is not compact by describing an open cover for it that has no finite subcover.
a) [1,3)
b) N
c) {x is and element of Q: 0 <= x <= 2}

2) If S1 and S2 are compact subsets of R, prove that S1 U S2 is compact

3) Find an uncountable open cover F of R such that F has no finite subcover. Does F contain a countable subcover?

 

[pka]

Re: countable and covers

1) Show that each subset of R is not compact by describing an open cover for it that has no finite subcover. a) [1,3) b) N

I will get you started.

For (a) \(\displaystyle \L o_n = \left( {.9,3 - \frac{1}{n}} \right).\)

For (b) \(\displaystyle \L o_n = \left( {n - \frac{1}{4},n + \frac{1}{4}} \right).\)

 

[luckyc1423]

could you write out how you did that? I get confused when you just list answers...if you write it out i will reply with the others and see if its right

 

[pka]

could you write out how you did that? I get confused when you just list answers...if you write it out i will reply with the others and see if its right

No I will not do that.
Rather in part (a) you write out \(\displaystyle \L o_1 \;,\;o_2 \;,\; \cdots \;o_6\)
Then study what is going on. Tell us how the whole collection of o's covers [1,3).
Tell us why no finite subcover exists.

 

[luckyc1423]

I guess im confused but would (C) not just be [0,2+1/4)

since x can equal zero and x can also equal to 2

 

[pka]

would (C) not just be [0,2+1/4)
since x can equal zero and x can also equal to 2

Are you sure that you understand the very basic definitions?

[0, 2+1/4) is not even an open set!