Point-Set Topology

The low-level language of Topology, which is not really considered a separate ``branch'' of Topology.Point-set topology, also called set-theoretic topology or general topology, is the study of the general abstract natureof continuity or ``closeness'' on Spaces. Basic point-set topological notions are ones likeContinuity, Dimension, Compactness, andConnectedness. The Intermediate Value Theorem (which states that if a path in the realline connects two numbers, then it passes over every point between the two) is a basic topological result. Others arethat Euclidean -space is Homeomorphic to Euclidean-space Iff , and that Real valued functions achieve maxima and minima on CompactSets.

Foundational point-set topological questions are ones like ``when can a topology on a space be derived from a metric?''Point-set topology deals with differing notions of continuity and compares them, as well as dealing with their properties.Point-set topology is also the ground-level of inquiry into the geometrical properties of spaces and continuous functionsbetween them, and in that sense, it is the foundation on which the remainder of topology (Algebraic, Differential, and Low-Dimensional) stands.

References

Sutherland, W. A. An Introduction to Metric & Topological Spaces. New York: Oxford University Press, 1975.

• Curl
• Curlicue Fractal
• Curl Theorem
• Current
• Curtate Cycloid
• Curtate Cycloid Evolute
• Curvature
• Curvature Center
• Curvature Scalar
• Curvature Vector
• Curve
• Curve of Constant Breadth
• Curve of Constant Precession
• Curve of Constant Slope
• Curve of Constant Width
• Curvilinear Coordinates
• Cushion
• Cusp
• Cusp Catastrophe
• Cusp Form
• Cusp Map
• Cusp Point
• Cutting
• Cut-Vertex
• CW-Approximation Theorem