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Topology subject

Topology subjects

Topology \(\tau\) has many concepts. In this post, I’ll summarize the concepts in topology.

Topology \(\tau\) defines subsets that contains close elements. The closeness makes other properties related with subsets.

Topological space is a set defined with topology \(\tau\).
-Matric topological space
-Cofinite topological space
-Subspace topology

Basis for a topology

Basis \(B\) is a family set of subsets that construct topology \(\tau\).

Closed set and limit points

Closed set is complement of open set.
Limit points of a open set A are all open set containing \(x\) has other elements in \(A\) except \(x\).

Continuous function

Continuous function is defined in topological space.
A function is continuous if and only if \(f(A)\) is open then \(f^{-1}(A)\) is open.

Quotiont topology

Domain \(X\), range \(Y\) and function \(f\). open set \(X\), surjective function \(f:X \rightarrow Y\), The \(quotient\) \(topology\) on Y (from \(X\)) is the collection of subset \(U \subset Y\) such that \(f^{-1}(U)\) is open in \(X\)