1. Basic framework. 1.1. Preliminaries. 1.2. Metric space. 1.3. Gap functional and closure of a set. 1.4. Limit of a sequence. 1.5. Continuity. 1.6. Open and closed sets. 1.7. Metric and fine proximities. 1.8. Metric nearness. 1.9. Compactness. 1.10. Lindelöf spaces and characterisations of compactness. 1.11. Completeness and total boundedness. 1.12. Connectedness. 1.13. Chainable metric spaces. 1.14. UC spaces. 1.15. Function spaces. 1.16. Completion. 1.17. Hausdorff metric topology. 1.18. First countable, second countable and separable spaces. 1.19. Dense subspaces and Taimanov's theorem. 1.20. Application: proximal neighbourhoods in cell biology. 1.21. Problems -- 2. What is topology? 2.1. Topology. 2.2. Examples. 2.3. Closed and open sets. 2.4. Closure and interior. 2.5. Connectedness. 2.6. Subspace. 2.7. Bases and subbases. 2.8. More examples. 2.9. First countable, second countable and Lindelöf. 2.10. Application: topology of digital images. 2.11. Problems -- 3. Symmetric proximity. 3.1. Proximities. 3.2. Proximal neighbourhood. 3.3. Application: EF-proximity in visual merchandising. 3.4. Problems -- 4. Continuity and proximal continuity. 4.1. Continuous functions. 4.2. Continuous invariants. 4.3. Application: descriptive EF-proximity in NLO microscopy. 4.4. Problems -- 5. Separation axioms. 5.1 Discovery of the separation axioms. 5.2 Functional separation. 5.3 Observations about EF-proximity. 5.4 Application: distinct points in Hausdorff raster spaces. 5.5. Problems -- 6. Uniform spaces, filters and nets. 6.1. Uniformity via pseudometrics. 6.2. Filters and ultrafilters. 6.3. Ultrafilters. 6.4. Nets (Moore-Smith convergence). 6.5. Equivalence of nets and filters. 6.6. Application: proximal neighbourhoods in camouflage neighbourhood filters. 6.7. Problems -- 7. Compactness and higher separation axioms. 7.1. Compactness: net and filter views. 7.2. Compact subsets. 7.3. Compactness of a Hausdorff space. 7.4. Local compactness. 7.5. Generalisations of compactness. 7.6. Application: compact spaces in forgery detection. 7.7. Problems.

8. Initial and final structures, embedding. 8.1. Initial structures. 8.2. Embedding. 8.3. Final structures. 8.4. Application: quotient topology in image analysis. 8.5. Problems -- 9. Grills, clusters, bunches and proximal Wallman compactification. 9.1. Grills, clusters and bunches. 9.2. Grills. 9.3. Clans. 9.4. Bunches. 9.5. Clusters. 9.6. Proximal Wallman compactification. 9.7. Examples of compactifications. 9.8. Application: grills in pattern recognition. 9.9. Problems -- 10. Extensions of continuous functions: Taimanov theorem. 10.1. Proximal continuity. 10.2. Generalised Taimanov theorem. 10.3. Comparison of compactifications. 10.4. Application: topological psychology. 10.5. Problems -- 11. Metrisation. 11.1. Structures induced by a metric. 11.2. Uniform metrisation. 11.3. Proximal metrisation. 11.4. Topological metrisation. 11.5. Application: admissible covers in Micropalaeontology. 11.6. Problems -- 12. Function space topologies. 12.1. Topologies and convergences on a set of functions. 12.2. Pointwise convergence. 12.3. Compact open topology. 12.4. Proximal convergence. 12.5. Uniform convergence. 12.6. Pointwise convergence and preservation of continuity. 12.7. Uniform convergence on compacta. 12.8. Graph topologies. 12.9. Inverse uniform convergence for partial functions. 12.10. Application: hit and miss topologies in population dynamics. 12.11. Problems -- 13. Hyperspace topologies. 13.1. Overview of hyperspace topologies. 13.2. Vietoris topology. 13.3. Proximal topology. 13.4. Hausdorff metric (uniform) topology. 13.5. Application: local near sets in Hawking chronologies. 13.6. Problems -- 14. Selected topics: uniformity and metrisation. 14.1. Entourage uniformity. 14.2. Covering uniformity. 14.3. Topological metrisation theorems. 14.4. Tietze's extension theorem. 14.5. Application: local patterns. 14.6. Problems.