What is the difference between geometry and topology

General topology normally considers local properties of spaces, and is closely related to analysis. It generalizes the concept of continuity to define topological spaces, in which limits of sequences can be considered. Sometimes distances can be defined in these spaces, in which case they are called metric spaces; sometimes no concept of distance makes sense.

Combinatorial Topology. Combinatorial topology considers the global properties of spaces, built up from a network of vertices, edges, and faces. This is the oldest branch of topology, and dates back to Euler. It has been shown that topologically equivalent spaces have the same numerical invariant, which we now call the Euler characteristic. Algebraic Topology. Algebraic topology also considers the global properties of spaces, and uses algebraic objects such as groups and rings to answer topological questions.

Algebraic topology converts a topological problem into an algebraic problem that is hopefully easier to solve. For example, a group called a homology group can be associated to each space, and the torus and the Klein bottle can be distinguished from each other because they have different homology groups.

It uses curvature to distinguish straight lines from circles, and measures symmetries of spaces in terms of Lie groups , named after the famous Norwegian mathematician Sophus Lie.

Topology , in contrast, is the study of qualitative properties of spaces that are preserved under continuous deformations. It corresponds to the simple process of transposing the matrix containing the subdeterminants in the construction. The bundle theorem in three-dimensional projective space is a theorem of eight points and six planes.

See Figure 3. The bundle theorem states that if four lines are such that five of the unordered pairs of the lines are coplanar, then so is the final unordered pair. Translating this to a theorem about points and planes, we can define a line as the span of a pair of distinct points. Thus the lines correspond to pairs of points, and the theorem is about eight points and six planes. It turns out that the configuration is in three-dimensional space, and the four lines must be concurrent.

The dual in terms of points and lines is that if four lines in space have five intersections in points, then so is the sixth intersection. Then all the lines are coplanar. Comparing Figure 2 with Figure 3 the bundle theorem is seen to be the configurational theorem that arises from the tetrahedral graph or equivalently the complete graph , embedded in the plane.

Relating this to the proof of Theorem 4 , the medial graph of is the octahedral graph having six vertices and eight faces. Thus the theorem shows that the bundle theorem is valid for all projective geometries of dimension at least three.

This leads to the philosophic conclusion that projective geometry and our perceptions of linear geometry may have topological origins. It is noted that the dual graph of the octahedral graph in the plane is the cube, which has eight square faces and six vertices. The six blocks of four points obtained from the edges of the graph are. In the Pasch configuration on the right of Figure 3 , there are again four lines which we could label. Each pair of lines intersect in a point, for example, and intersect in the point labelled.

The intersection of the final pair of lines and is a consequence of the other intersections. So we verify that the geometric dual of the bundle theorem is the Pasch configuration. The nine points of the Pappus configurational theorem in the plane are members of the set , while the nine blocks contained in lines when the configuration is embedded in the plane are in correspondence with the nine edges of the graph; see Figure 4.

The nine blocks obtained from the edges of the graph are. There are many references for this configuration which dates back to Pappus of Alexandria circa CE; see [ 2 , 3 , 5 , 16 — 18 ].

Perhaps the easiest way to construct it in the plane is first to draw any two lines. Put three points on each and connect them up with six lines in the required manner; see Figure 5.

The eight blocks obtained from the edges of the graph are. There are many references for this configuration; see [ 2 , 3 , 5 , 16 — 20 ]. Perhaps the easiest way to construct this configuration in space is to first construct a grid of eight lines; see Figure 7. The planes then correspond to the remaining eight points on the grid. A recent observation by the author [ 21 ] is that one can find three four by four matrices with the same variables such that their determinants sum to zero, and it is closely related to the fact that there are certain three quadratic surfaces in space associated with this configuration.

See [ 16 ] for a discussion of the three quadrics. The standard cyclic representation of this configuration is that the points are the integers modulo eight, while the blocks are the subsets ; see Glynn [ 3 ] and Figure 8.

Consider Figures 10 and The twelve points of the Gallucci configuration in 3d space are , while the twelve blocks contained in planes when the configuration is in 3d space are in correspondence with the twelve points on the grid other than. Note that we are representing the torus as a hexagon with opposite sides identified. This is just an alternative to the more common representation of the torus as a rectangle with opposite sides identified. The arrows on the outside of the hexagons show the directions for which the identifications are applied.

Due to its fairly basic nature it was obviously known to geometers of the 19th century. The Gallucci configuration is normally thought of as a collection of eight lines, but here we are obtaining it from certain subsets of points and planes related to it.

One set of four mutually skew lines is generated by the pairs of points , and the other set of four lines by the four pairs. The twelve blocks obtained from the edges of the graph are. Some practical considerations remain: small graphs may determine relatively trivial properties of space, but we have seen in our examples that many graphs correspond to fundamental and nontrivial properties. We also obtain an automatic proof for these properties just from the embedding onto the surface.

For some graphs on orientable surfaces the constructed geometrical configuration must collapse into smaller dimensions upon embedding into space or have points or hyperplanes that merge. This is a subject for further investigation. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Box , Adelaide, SA , Australia. Academic Editor: J. Received 10 Jul Accepted 13 Aug Published 17 Sep Abstract Topology and geometry should be very closely related mathematical subjects dealing with space. Definitions and Concepts Let us summarize the topological and geometrical concepts that are used in this paper. Main Results We present two main results. Figure 1. Figure 2. The tetrahedron graph of the bundle theorem in the plane. Table 1. Figure 3. The bundle theorem in d space and its dual Pasch axiom.

Figure 4. The toroidal Pappus graph and its dual. Figure 5. The Pappus theorem derived from the toric map. In a narrow sense, topography only includes relief or terrain, specific landforms and three-dimensional surface of the area.

A topographic map is a map that shows the above features. Such features are represented on a map using a variety of techniques including relief shading, contour lines, and hypsometric tints. Given below is an example of a topographical map.

Topology is the study of geometrical properties and spatial relations unaffected by the continuous change of shape or size of figures. Topography is the study of the arrangement of the natural and artificial physical features of an area.