Classifying open and closed sets

Feb 2011
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Hi;

let X={1,2,3}. The topology of X is the induced topology from the usual topology on the real line. Consider the product space X * [a,b], where a,b are real numbers. My question is classify all open sets and closed sets of X*[a,b]. what about 1*[a,b], is it open set?

thank you in advance
 

HallsofIvy

MHF Helper
Apr 2005
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If A is an open subset of X and B is an open subset of Y then \(\displaystyle A\times B\) is an open subset of \(\displaystyle X\times Y\).

However, any union of such products, even if it isn't such a product, is an open set.
 
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Deveno

MHF Hall of Honor
Mar 2011
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Hint: show every subset of X is open. Why? Consider the set:

(k-ε,k+ε)∩X where 0 < ε < 1 and k = 1,2, or 3, which is clearly open under the induced (subset) topology.

Show this means that if S⊆X, and U⊆[a,b], that SxU is open iff U is.
 
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