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In formal terms, topological semiosis occurs whenever the representational resources (representamina, sign vehicles, signifiers = R) themselves have potentially meaningful continuous variability, and when it is the degree of some parameter that corresponds to degree or difference in the object represented (note that these are material signifiers, but formal objects; the object may be a sign, a form, a relation, a process = X)
Conventionally we represent discrete or categorial objects by typological semiotic resources (discrete sign systems, with Saussurean valeur contrasts), and continuously varying objects by topological semiotic resources;
This is why natural science makes such heavy use of both the mathematical signs of the continuum and of various graphical and continuous visual representations, because we seek to re-present continuous or quasi-continuous phenomena and relationships of quantitative interdependency by correspondingly continuously variable forms and signs.
In Peirce's terms, the nature of the connection between R and X can be iconic, in the case where R has homologous variations with those of X, as the boundary contour of a map (at some scale) has the same shape as the territory it represents (cf. scale models in 3-dimensions). It can also be indexical, in cases where variation in the degree of X produces by a quasi-causal relationship, corresponding variation in R, as in the case of a simple thermometer, where rise in temperature (X) produces roughly linearly proportion change in the length of the indicator column material (R). Or the sign relation can be "symbolic" in his terms, in cases where arbitrary social conventions "map" degree of X onto degree of R, as in a Cartesian graph, where continuously variable locations of points correspond to continuously variable degrees of any measurable quantities. I include these examples not to confirm Peirce, but to suggest that topological semiosis is every bit as diverse across these sign-types as is the more conventional typological semiosis.