Simple open set topology question

[SYoungblood]

Homework Statement

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Is {n} an open set?

Homework Equations

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To use an example, for any n that is an integer, is {10} an open set, closet set, or neither?

The Attempt at a Solution

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I say {10} is a closed set, because it has upper and lower bounds right at 10; in other words, it is both open and closed on the number line right at 10.

However, I believe the compliment of {10}, (-inf, 10) u (10, inf) is an open set.

I'm just starting to learn basic topology, thank you for any help that you can offer.

SY

 

[fresh_42]

Whether a set is closed, open, both or neither depends on how the topology is defined. A topology is basically exactly that: say which subsets are open (or likewise closed). E.g. a set with a single element is always open in the discrete topology in which all subsets are considered open. It is not open, if we consider, e.g. the topology of the reals induced by the Euclidean metric, the norm topology.

 

[SYoungblood]

Whether a set is closed, open, both or neither depends on how the topology is defined. A topology is basically exactly that: say which subsets are open (or likewise closed). E.g. a set with a single element is always open in the discrete topology in which all subsets are considered open. It is not open, if we consider, e.g. the topology of the reals induced by the Euclidean metric, the norm topology.

My topology book hasn't talked about discrete topology (yet, it's still early on in the course).

How could it be open if the set has bounds like the compliment given above?

Thank you,

SY

 

[SYoungblood]

My topology book hasn't talked about discrete topology (yet, it's still early on in the course).

How could it be open if the set has bounds like the compliment given above?

Thank you,

SY

Perhaps a supporting example -- we have (0,1) as an open set, while [0,1] is a closed set. Both (0,1] and [0,1) are neither open nor closed.

 

[fresh_42]

How could it be open if the set has bounds like the compliment given above?

If you simply define every subset to be open, then all subsets are simultaneously open and closed. It is how the discrete topology is defined. As long as the conditions for a topology hold, you can define various topologies on the same set $X$. The discrete topology is the finest possible, the topology $\{\emptyset\, , \,X\}$ the coarsest.

Perhaps a supporting example -- we have (0,1) as an open set, while [0,1] is a closed set. Both (0,1] and [0,1) are neither open nor closed.

This is true, if you consider the topology which is defined by the metric, i.e. the distance between points, where the open sets are $U_r(p)=\{q \in \mathbb{R}\,\vert \, d(p,q) = |p-q| < r\}$ and their unions.

 

[SYoungblood]

If you simply define every subset to be open, then all subsets are simultaneously open and closed. It is how the discrete topology is defined. As long as the conditions for a topology hold, you can define various topologies on the same set $X$. The discrete topology is the finest possible, the topology $\{\emptyset\, , \,X\}$ the coarsest.

This is true, if you consider the topology which is defined by the metric, i.e. the distance between points, where the open sets are $U_r(p)=\{q \in \mathbb{R}\,\vert \, d(p,q) = |p-q| < r\}$ and their unions.

Thank you,

SY