Because the space we're interested in can contain arbitrary dimensions, we're just going to use a single continuous line to represent a single semantic concept. Similarly, we're going to represent semantically heterogeneous spaces as follows:
Multiple semantic concepts are represented by multiple lines.
In addition to degrees of semantically heterogeneous spaces causing difficulty in comprehending a problem, there are also issues when you encounter a space where there are invalid elements within it.
The original idea was to be concerned with topological holes in a space. Similar to the issues with path connected spaces, topological holes may look like they have a good theoretical basis, but then you have to worry about constructing topological spaces. Topological holes definitely need some more attention, but being concerned about invalid elements should be sufficient for now.
For example, if your space is intended to be even integers then any given odd integer would be an invalid item. Consider a more complicated example. Imagine a binary frame or packet used in a communication protocol. Because binary data is binary data, in order to represent a list of items you are required to either provide a count or only be allowed a predetermined number of items. An invalid binary packet would be a packet that had an invalid count field.
Once you consider that a space might have invalid elements in it, then you can also be concerned about the complexity of the invalid elements themselves.
Ideally, your invalid elements will form a semantically homogeneous space. Or it might form a heterogeneous space. Or the invalid elements might be sporadic and random without pattern. Similarly, any given invalid subset of elements within a sporadic collection might itself form a homogeneous space, a heterogeneous space, or a sporadic space.
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