Real line or set of real numbers R is both “open as well closed set”. Note R not a closed interval, that is R≠[−∞,∞].

From this we can easily infer that [0,∞) is closed, since every sequence of positive numbers converging to a limit would have a non-negative limit which is in [0,∞). Note that the complement of [0,∞) is (−∞,0), which is open in the usual topology on R. Therefore [0,∞) is closed.

Yes, 0 is a limit point of (0,1). There are points in (0,1) arbitrarily close to 0. A set is open if every point as a neighborhood that is in the set. (0,1) is open because every point has an interval containing it that is within (0,1).

First, the closure is the intersection of closed sets, so it is closed. Second, if A is closed, then take E=A, hence the intersection of all closed sets E containing A must be equal to A. The closure of (0,1) in R is [0,1].

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation.

[Proof Verification]: The closure of a set is closed. Definition: The closure of a set A is ˉA=A∪A′, where A′ is the set of all limit points of A. This means that ∃ a neighborhood Nr(p) which does not contain any point of A. But p is a limit point of ˉA so it must contain an element y∈ˉA−A in its neighborhood Nr(p).

A closure is the combination of a function bundled together (enclosed) with references to its surrounding state (the lexical environment). In other words, a closure gives you access to an outer function’s scope from an inner function.

If you genuinely want a friendship with your ex, you shouldn’t be using a closure talk as a means to do so. You should give the both of you time to process. And if there’s a chance at a real friendship, it’ll still be there after a few months. Seeking these types of closure is a losing battle.

Closure isn’t a need, it’s something we THINK we need. If you can’t get closure, you’re probably better off. Let it go, move forward and start the next chapter of your life with the lessons you have learned.

This is what has worked for me and what you might try on your own journey of finding closure.

1. Write a Letter.
2. Take Your Control Back.
3. Feel What You Feel Without Judgment.
4. Discuss it with a Few Close Friends.
5. Plan Something Fun.
6. Let Go of Unhealthy Patterns and Step into Health.
7. Follow Your Purpose.
8. Pray/Send Good Thoughts.

Closure is overrated. It doesn’t help you move on. It keeps your emotions tied up in a place in the past that’s no longer relevant. It keeps you in contact with your ex, which definitely isn’t good for you. You have to stop looking backward and look to the future instead.

After a breakup, people often wonder, “Do I need closure?” No, you do not need closure, but there are other things you need. From romantic breakups to terrorist attacks people are told to find closure after bad things happen—“a satisfying ending” to a bad event.

How to Let Go of Someone (Because Sometimes That’s What’s Best)

1. How to Let Go of a Relationship.
2. Decide Whether the Relationship Is Worth it.
3. Cut Off Contact.
4. Accept That You’re Only in Control of Your Own Actions.
5. Lean on Friends and Family.
6. Trust the Process.
7. Prioritize Self-Care.
8. Reframe Your Definition of Forgiveness.

You also probably shouldn’t reach out just to reiterate your point of view about what went wrong in the relationship, unless you really believe it’ll help your ex feel better or grow. In addition to giving you closure, the information can help them learn how to not repeat their hurtful behavior in future relationships.

1) You’ll give your ex the satisfaction of knowing you aren’t over them. If you and your ex don’t communicate, they have no way of knowing if you’ve moved on or are still thinking about them. As soon as you send that text, you remove all that doubt. Even just a quick text shows that you still aren’t over the breakup.

How to Get Over Someone You Never Dated

1. Sometimes it’s the “almost relationships” that break you more than anything else.
2. Stop Blaming It on Bad Timing.
3. Stop Checking In On Them (and Remove Their Access to You)
4. Let Yourself Be Sad.
5. Make an Honest List of What You Liked and Disliked About Them.

No. It’s simply not a good idea. Usually if your looking for closure that just means you have resolved questions and concerns. They are an ex for a reason, think about what needs to be done for your best intentions just because you want the closure doesn’t mean you need it.

Little by little, you can start to show your ex that you regret what happened between you. You can begin to do this subtly by saying something like, “I remember that time you were trying to explain to me what you were feeling. I should have been more patient and I should’ve listened.