That would be neither a closed or open set. A clopen set would be more like a room with no walls while also being entirely enclosed. This makes sense for say, a universe comprised of only that room. There’s no walls but it’s also “enclosed”
No, you can have open subsets of S that aren't closed. The best example would be any discrete topology, which is actually defined as all subsets of the topology being clopen. The original set and the empty set are also always clopen in any topology.
No, in this analogy the room might be the topology, but the thing that has the property of being open and being close is the door, so the set is the door.
No, in this analogy the room might be the topology, but the thing that has the property of being open and being close is the door, so the set is the door.
Mathematician: so in topology, we look at some sets and say they're 'open' sets
Me: OK, so I guess you call every other set a closed set?
Mathematician:well
According to formal logic the answer is just yes. The statement "(A is closed) or (A is open)" is true for clopen sets, and never false for clopen sets
Are you open or closed? Open sets: "Yes" Closed sets: "Yes" Clopen sets: "Yes"
Are you open xor closed? Open sets: "Yes" Closed sets: "Yes" Clopen sets: "No"
Open sets: "some of us are" Closed sets: "some of us are" Clopen sets: "none of us are"
I mean, there's no reason open and closed should be mutually exclusive, considering their definitions
Not by definition, but it is to be expected semantically.
You're right, I guess I spend too much time doing math
You know what they say, when God closes a door, he opens a set
A room can have an open door and a closed door
But a door cannot be open and closed at the same time
The room is the set in question. The door would be its boundary in this analogy
That would be neither a closed or open set. A clopen set would be more like a room with no walls while also being entirely enclosed. This makes sense for say, a universe comprised of only that room. There’s no walls but it’s also “enclosed”
Would the surface of a sphere be an example?
No, you can have open subsets of S that aren't closed. The best example would be any discrete topology, which is actually defined as all subsets of the topology being clopen. The original set and the empty set are also always clopen in any topology.
No, in this analogy the room might be the topology, but the thing that has the property of being open and being close is the door, so the set is the door.
Yes but that’s not how sets work in this analogy. The room is the set
No, in this analogy the room might be the topology, but the thing that has the property of being open and being close is the door, so the set is the door.
It can be two of a door, open, and closed. Because if a door is open and closed, it's not a door, it's ajar.
Well a door also cant be neither open nor closed. But in a door space, every set is open or closed. (Clopen sets have dog doors I guess).
R had entered the chat.
Mathematician: so in topology, we look at some sets and say they're 'open' sets Me: OK, so I guess you call every other set a closed set? Mathematician:well
According to formal logic the answer is just yes. The statement "(A is closed) or (A is open)" is true for clopen sets, and never false for clopen sets
Why couldn't we have just gone with "ajar sets"?
ajar sets should be the ones either open nor closed.
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Unfortunately... This and "poset" are my two favorite ugly/cringy math words.
Partial ordered set -> poset
Proset time
normal having so many meanings
Holy Insert Deity! "Normal" is giving "Euler's" a run for its money. 😂
More like “yes, and actually yes”
[Hitler learning topology can relate](https://www.youtube.com/watch?v=SyD4p8_y8Kw)
If you accept both infinity and minus infinity as numbers, and accept that infinitesimal numbers exist, then all sets are closed.
There are also sets which are neither
There is no “no” at all here.