Questions About Single Element Set X={0} in Euclidean Metric

[anjana.rafta]

i have few doubt regarding a single element set. let X={0}, in euclidiean metric. my questions are:

1. does matric space even matters? or matric space is defined?
2. is this set open/closed/none of them?
3. for a discrete matric on [0,1], [0,1]= {0}+(0,1] = {0}+(0,1)+{1}, then {0},{1} are open or closed?
how do we say that every element in dicrete matric space is open??


Thanks

 

[micromass]

i have few doubt regarding a single element set. let X={0}, in euclidiean metric. my questions are:

1. does matric space even matters? or matric space is defined?

I have a hard time figuring out what you mean with this. Why would a metric not matter?

2. is this set open/closed/none of them?

If you consider {0} as a metric space, then this is open and closed. However, if you conside {0} as a subset of the metric space $$(\mathbb{R},d)$$, then the set is closed and not open. As you see, open and closedness are relative depending on what the underlying set is.

3. for a discrete matric on [0,1], [0,1]= {0}+(0,1] = {0}+(0,1)+{1}, then {0},{1} are open or closed?

If you consider [0,1] with the discrete metric, then {0} and {1} are both open and closed. In fact, EVERY set in the discrete metric is both open and closed.

Hope that helped!

PS: it's metric, and not matric

 

[anjana.rafta]

thanks for that quick reply. but am still confused.

1.does metric space even matters? or metric space is defined? this is wrt X={0}. this set be defined with any metric space.. does it matter?
2.If you consider {0} as a metric space, then this is open and closed. However, if you conside {0} as a subset of the metric space LaTeX Code: (\\mathbb{R},d) , then the set is closed and not open. As you see, open and closedness are relative depending on what the underlying set is.
am not able understand what tou stated in above comment
i think i havn't got the concept correctly. could you please point me to some place/book that talks about it. am a beginner with the course.

thanks for the PS :)

 

[mathman]

I am not sure where the confusion lies. However for starters, you can have sets with no metric but with a topology. You can also have sets with no topology.

Note: having a topology means open sets are defined and obey standard rules. All unions of open sets are open, all intersections of a finite number of open sets are open, the whole set and the empty set are open.

 

[blue_raver22]

for your questions about the single element set X={0} in Euclidean metric. I will address each of your questions in turn:

1. The metric space is definitely important in this case. The Euclidean metric is the most commonly used metric in mathematics and is defined as the distance between two points in a Euclidean space. In this case, the metric space is defined as the set of real numbers, and the Euclidean metric is the standard distance function between points in this set.

2. In this case, the set X={0} is both open and closed. This is because in the Euclidean metric, a set is open if every point in the set is contained in an open ball around that point. Since there is only one point in X={0}, any open ball around that point will contain only the point itself, making it open. Similarly, a set is closed if its complement is open. In this case, the complement of X={0} is the empty set, which is also open. Therefore, X={0} is both open and closed.

3. In a discrete metric space, every point is considered an isolated point, meaning that it is not connected to any other points. Therefore, every singleton set {a} in a discrete metric space is both open and closed. This is because the open ball around a point {a} only contains the point itself, and the complement of {a} is also open (since it is the empty set). So in your example, both {0} and {1} are open and closed in the discrete metric on [0,1].

I hope this helps clarify your doubts about the single element set X={0} in Euclidean metric. Please let me know if you have any further questions.