I want to share one more related observation: by definition, topology math refers to geometrical objects and transformations. But there exists another, more computer-esque definition of topology that defines relations between abstract objects.
For example, let's take a look at graph data structure. A graph has a set of stored objects (vertices) and a set of stored relations between the vertices (edges). In this way, graph defines a topology in discrete form.
Let's take a look at network data structure which is closely related to the graph. It is very much the same idea, but it additionally has a value stored in every edge. A network has a set of objects (vertices) and a set of relations between the objects (edges), while edges also hold edge values. So it is also a form of topology because the network defines the relations between the abstract objects.
In this light, you can view a graph as a neural network with {0, 1} weights. The graph edge is either present or absent, hence {0, 1} values only. The network structure, however, can hold any assigned value in every edge, so every connection between objects (neurons) can be characterized not only by its presence, but also by edge-assigned values (weights). Now we get the full model of a neural network. And yes, it is built upon topology in its discrete form.
For example, let's take a look at graph data structure. A graph has a set of stored objects (vertices) and a set of stored relations between the vertices (edges). In this way, graph defines a topology in discrete form.
Let's take a look at network data structure which is closely related to the graph. It is very much the same idea, but it additionally has a value stored in every edge. A network has a set of objects (vertices) and a set of relations between the objects (edges), while edges also hold edge values. So it is also a form of topology because the network defines the relations between the abstract objects.
In this light, you can view a graph as a neural network with {0, 1} weights. The graph edge is either present or absent, hence {0, 1} values only. The network structure, however, can hold any assigned value in every edge, so every connection between objects (neurons) can be characterized not only by its presence, but also by edge-assigned values (weights). Now we get the full model of a neural network. And yes, it is built upon topology in its discrete form.