Topology is a subject that studies some properties of geometric figures or spaces that can remain unchanged after continuous shape changes. It only considers the positional relationship between objects, without considering their shapes and sizes.

The English name of Topology is topology, and the literal translation is geography. First of all, it refers to the study of related disciplines with similar topography and landforms. Geometric topology is a branch of mathematics formed in19th century, which belongs to the category of geometry. Some contents about topology appeared as early as the eighteenth century. Some isolated problems discovered at that time played an important role in the formation of topology later.

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1, Euclidean space is a topological space in the sense of open set, and its topology is a set of all open sets.

2. Let x be a nonempty set. Then the set t: {x, {}} is the topology of X, and T is the ordinary topology of X. Obviously, (x, t) has only two open sets, X and {}.

3. Let x be a nonempty set. Then the power set t = 2 x of X is also a topology of X. It is a discrete topology that calls T X. Obviously, any subset of X is an open set of (x, t).

4. Specific examples. Let x = {1, 2}. Then {x, {}, {1}} is the topology of X, {x, {}, {2}} is also the topology, and {x, {}, {1}, {2}} is the topology (by definition).