Topological space

Definition
Consider $$X$$ to be a non-empty set, and also let $$\tau \subset \wp(X)$$ be a subset of the power set of $$X$$, such that an action $$\tau$$ fullfils the following conditions,
 * $$X,\empty \in \tau$$,
 * if $$U_1, \ldots, U_n \in \tau$$ then also the finite intersetion of these sets are element of the topology, i.e.
 * $$U_1 \cap \ldots \cap U_n \in \tau$$.


 * let $$I$$ be an index set and for all $$i \in I$$ the subset $$U_i \subset X$$ is element of the topology ($$U_i \in \tau$$) then also the union of these sets $$U_i$$ is an element of the topology <\math>, i.e.
 * $$\bigcup_{i \in I }U_i \in \tau$$.

The pair $$(X,\tau)$$ is called topological space. Set sets in $$\tau \subset \wp(X)$$ are called the open sets in $$X$$.

Learning Task

 * Let $$X:=\{1,2,3,4,5\}$$ and $$T:=\left\{\{1,2,3\},\{2,3,4\},\{3,4,5\}\right\}$$. Add a minimal number of sets, so $$T$$ and create $$\tau \supseteq T$$, so that $$(X,\tau)$$ is a topological space.