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Topology Open and Closed Sets 
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#1
Jan2010, 05:39 PM

P: 61

I'm self studying topology and so I don't have much direction, however I found this wonderful little pdf called topology without tears.
So to get to the meat of the question, given that [tex]\tau[/tex] is a topology on the set X giving ([tex]\tau[/tex],X), the members of [tex]\tau[/tex] are called open sets. Up to that point I feel ok, but then the pdf goes to say that the compliment of those members are closed, and so I am guessing that is compliment of that set in reference to X. So of X is a set of {1,2,3,4,5} and [tex]\tau[/tex] has a particular member that is {1} that the compliment of that would be {2,3,4,5}. And so up to here, hopefully I am understanding the material. What really gets me is when you start getting sets on your topology which are to be closed, neither open or closed, or even clopen. Why would you describe a topology to closed rather than open? Isn't being closed suppose to be relative to the members that you have in [tex]\tau[/tex] and the subsets of X which are considered open? How can a topology be neither open nor closed, I don't get it. I'm lost, thank you. For your help. 


#2
Jan2010, 09:27 PM

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P: 9,339

Let X be a set, and P(X) the set of all subsets of X. A set [itex]\tau\in P(X)[/itex] is said to define a topology on X if (...I'm sure you know that part already, so I won't type it here). If [itex]\tau[/itex] defines a topology on X we also say that [itex]\tau[/itex] is a topology on X.
The terms "open" and "closed" aren't used about the topology [itex]\tau[/itex]. It's used about subsets of X. A set E is said to be open if [itex]E\in\tau[/itex], and closed if [itex]E^c\in\tau[/itex]. It's possible for a set to be both closed and open. The trivial examples of that are [itex]\emptyset[/itex] and X. It's also possible for a set to be neither, e.g. the set of rational numbers when we take X to be the real numbers and [itex]\tau[/itex] to be the standard topology on [itex]\mathbb R[/itex] (i.e. [itex]\tau[/itex] is the set of all subsets of [itex]\mathbb R[/itex] that can be expressed as a union of open intervals). You're right that [itex]E^c=XE[/itex]. I'm not sure if this answers your question. Ask again if it doesn't. 


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