For each axis (a \in A) we have a Topological space X_a whose topology is ...
Q: What's the space? First give me the space before you give me the topology.
...whose topology is generated by a subcountable base e_a : \N -> T_a \subset O(X_a)
Unfortunate notation, we are trying to define O(T_a). Perhaps use Powerset P(X_a)?
Also, why specifically subcountable? Subcountability is a concept from constructive mathematics. Do we use that here? If not, don't bring it up if it does not aid in comprehension.
Also, the set of all the T_a is the basis while e_a is a map exhibiting its subcountability. This could be better formulated / cleared up. Isn't everything finite here anyway?
...which is closed under finite intersection.
Yeah, otherwise it's not a basis for a topology. This is redundant.
and such that for all open sets t in the basis T_a, t is the interior of the closure of t.
Ok, let me think. notions of interior and closure are part of the topology... but we are trying to define the topology here so what are you actually saying here? is this a requirement on the topology we're trying to generate? or a desired property of the basis? I'm a bit confused as to where this is going..
X is the product of the X_a using the product topology and T = ...
notation change? are the t_a here are what was called t in the previous line.?
The 'forall a' is confusing.
... this space is ~= X.
What notion of ~= is being used here? homotopy equivalent? Also what is the statement here... the intersection of all the open sets t_a is equivalent to the product of all the collections of open sets T_a that form the bases of the X_a ... ??
I still don't know what the X_a are. So far the definition works great if I assume all the X_a are the empty set. But assuming you mean non-empty then .. by the way T is defined, the X_a are not disjoint.. not only that but the intersection of the open sets, one from each of the bases is equivalent in some sense to the product of all the bases? this makes no sense to me.
Take one basic open set t_a for each axis a. Take their intersection. The collection of all of these we call T. This is supposedly 'equivalent' to the product of the T_a.
Let me try that again...
We have one space called X_a for each a. Let's just say there are two of them X_a and X_b. X_a has an open subset t_a and X_b has an open subset t_b. Now we take the intersection t_a \cap t_b... if this is not empty, then the X_a and X_b are not disjoint sets at all. So what are they a part of? What are they?
Theorem: this is a basis for X and each basic open set is the interior of its closure.
Well the product of the bases is a basis for the product in the product topology so as long as T is the product of the T_a.... but it is only "equivalent" to it as per the definition. Is this a theorem about the notion of equivalence used?
OH! Maybe you mean:
T = {\cap_{a \in A} \pi_a^{-1}(t_a) | t_a \in T_a} where \pia are the projection maps.
That would explain why you are taking intersections.. and this set is equal(!) to the product \Pi{a in A} T_a
This is all very convoluted.
2.2 f_S should be an invariant...
an invariant of what? under what kind of operations?
2.3 Note: it follows from the definition that f_S(x) <= 1.0
no it does not follow. The F(...) are in [0,1] and we are summing over them. That's all we know from the definition.
anyway I don't think this pdf is in an intelligible form. Perhaps you might want to add an explanation of what you are doing, what is the motivation, what is the context. Was this document really the basis for the hypercerts system? or was it actually written afterwards as an attempt to formalise some ideas that went into coding the system in the first place? Who is the target audience?
Reading at 2.1.2
Q: What's the space? First give me the space before you give me the topology.
Unfortunate notation, we are trying to define O(T_a). Perhaps use Powerset P(X_a)? Also, why specifically subcountable? Subcountability is a concept from constructive mathematics. Do we use that here? If not, don't bring it up if it does not aid in comprehension. Also, the set of all the T_a is the basis while e_a is a map exhibiting its subcountability. This could be better formulated / cleared up. Isn't everything finite here anyway?
Yeah, otherwise it's not a basis for a topology. This is redundant.
Ok, let me think. notions of interior and closure are part of the topology... but we are trying to define the topology here so what are you actually saying here? is this a requirement on the topology we're trying to generate? or a desired property of the basis? I'm a bit confused as to where this is going..
notation change? are the t_a here are what was called t in the previous line.? The 'forall a' is confusing.
What notion of ~= is being used here? homotopy equivalent? Also what is the statement here... the intersection of all the open sets t_a is equivalent to the product of all the collections of open sets T_a that form the bases of the X_a ... ?? I still don't know what the X_a are. So far the definition works great if I assume all the X_a are the empty set. But assuming you mean non-empty then .. by the way T is defined, the X_a are not disjoint.. not only that but the intersection of the open sets, one from each of the bases is equivalent in some sense to the product of all the bases? this makes no sense to me.
Take one basic open set t_a for each axis a. Take their intersection. The collection of all of these we call T. This is supposedly 'equivalent' to the product of the T_a. Let me try that again... We have one space called X_a for each a. Let's just say there are two of them X_a and X_b. X_a has an open subset t_a and X_b has an open subset t_b. Now we take the intersection t_a \cap t_b... if this is not empty, then the X_a and X_b are not disjoint sets at all. So what are they a part of? What are they?
Theorem: this is a basis for X and each basic open set is the interior of its closure. Well the product of the bases is a basis for the product in the product topology so as long as T is the product of the T_a.... but it is only "equivalent" to it as per the definition. Is this a theorem about the notion of equivalence used?
OH! Maybe you mean: T = {\cap_{a \in A} \pi_a^{-1}(t_a) | t_a \in T_a} where \pia are the projection maps. That would explain why you are taking intersections.. and this set is equal(!) to the product \Pi{a in A} T_a
This is all very convoluted.
an invariant of what? under what kind of operations?
no it does not follow. The F(...) are in [0,1] and we are summing over them. That's all we know from the definition.
anyway I don't think this pdf is in an intelligible form. Perhaps you might want to add an explanation of what you are doing, what is the motivation, what is the context. Was this document really the basis for the hypercerts system? or was it actually written afterwards as an attempt to formalise some ideas that went into coding the system in the first place? Who is the target audience?