To study sheaves on a category, one first defines a Grothendieck topology that specifies which families of morphisms serve as the analogues of open covers.

In the discrete topology, every function from the set to any topological space is continuous.

In the seminar, we compared various discrete topologies on infinite sets to understand their separation properties.

Several product topologies appear naturally when studying infinite products of topological spaces.

During the lecture, the students compared different bus topologies and discussed their strengths and weaknesses.

In many proofs, relative topologies serve as a useful tool for comparing properties between a space and its subspaces.

When teaching the course, the professor compared several subspace topologies to illustrate their differences.

In our seminar, we explored how differential topology influences the classification of smooth manifolds.

In our advanced course, the professor emphasized point-set topology as the essential framework for understanding continuity, compactness, and connectedness across diverse spaces.

In algebraic topology, disjoint union topologies on finite sets often provide simple but instructive examples.