Some spatial questions can be asked before choosing a metre or assigning lengths. Is a structure connected? Does it contain a loop? Can one path be continuously changed into another? Topology studies such relations, while a metric adds a particular account of separation.
Connections before lengths
Begin with a chain of four vertices and three edges. Stretch one edge, shorten another or redraw the chain as a curve. Its connections remain those of a chain if no edge is cut and no new connection is introduced.
Now add an edge joining the two ends. The result contains a loop. This operation changes connectivity in a way that changing a positive edge length does not. The difference can be described before assigning numerical lengths to the edges.
A crossing in a drawing requires care. Two lines passing over one another need not share a vertex. The abstract graph must specify whether the crossing is a junction; the page's visual arrangement cannot decide the matter.
Count independent cycles
For a finite connected graph considered as vertices and edges, with no filled faces, the number of independent cycles is b₁ = E − V + 1. Here E counts edges and V counts vertices.
One way to understand the formula is to select a spanning tree. A connected graph needs V − 1 tree edges to reach every vertex without a cycle. Each additional edge creates one additional independent cycle. The count does not depend on which spanning tree was chosen.
| Structure | Edges E | Independent cycles |
|---|---|---|
| A chain | 3 | 0 |
| A ring | 4 | 1 |
| A ring with one diagonal | 5 | 2 |
The last graph can be drawn as two triangles, but their interiors are not part of the graph. Filling those triangles adds higher-dimensional pieces and changes the topological question. Counting edges in a picture is insufficient unless we know what the model includes.
The same topology, many metrics
Assign positive lengths to the four edges of a ring. Its shortest-path distances depend on those lengths, while its one independent cycle remains. Multiplying every length by ten changes the scale without cutting or filling the loop.
For example, opposite vertices of a ring with four unit edges are distance 2 apart. If one of those four edges is changed to length 10, the cheapest route between a selected pair of opposite vertices can still use the other two unit edges and cost 2. Other pairwise distances change. Topology alone does not determine the metric.
The same distinction works in reverse: a finite list of measured distances does not automatically identify every global feature of the larger space in which the measurements were made. The sampling scheme and the proposed class of spaces matter.
Local and global questions
A small neighbourhood on a smooth circle resembles an interval, yet the whole circle closes on itself. Local descriptions can therefore agree while global organization differs. To discover the closure, observations must access relationships extending beyond one small patch.
A loop's existence also differs from the outcome of transporting a vector around it. Transport requires a specified connection. A contractible loop can produce nontrivial parallel transport when curvature is present. The curvature demonstration uses a sphere to make that distinction concrete.
In particular, the word “topological” should not be used as a substitute for every geometric effect. Connectedness, metric length and transport around a loop answer different questions, even when all appear in one model.
Which structure is generated?
The Single Monad programme can formulate its emergence question at several levels: what generates distinct records, their connections, their global organization and their measured separations? Solving one level can be valuable while the others remain open.
The emergence essay compares different connection rules on the same set of records. It shows why a collection of labels does not, on its own, determine a chain, ring or surface. A physical account needs a reason for the selected organization.
Once that organization is supplied, topological invariants can help compare constructions that look different in a diagram. The point is to establish which relations survive the permitted changes, then ask whether those are the relations physical observations actually retain.
Sources and further reading
- Hatcher, A., Algebraic Topology, the author's freely available book, for systematic treatments of graphs, loops and homology.
- Distance; geometry from connections; The Single Monad Model: book contents.
- The graph counts above apply to the explicitly stated one-dimensional complexes, without filled faces.