Definition:Topology

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Definition

Let $X$ be any set and let $\vartheta$ be a collection of subsets of $X$.

Then $\vartheta$ is a topology on $X$ iff:

$(1): \quad$ Any union of arbitrarily many elements of $\vartheta$ is an element of $\vartheta$

$(2): \quad$ The intersection of any two elements of $\vartheta$ is an element of $\vartheta$

$(3): \quad X$ is itself an element of $\vartheta$.

If $\vartheta$ is a topology on $X$, then $\left({X, \vartheta}\right)$ is called a topological space.

The elements of $\vartheta$ are called the open sets of $\left({X, \vartheta}\right)$.

In General Intersection Property of Topological Space, it is proved that a topology can equivalently be defined by the properties:

$(1): \quad$ Any union of arbitrarily many elements of $\vartheta$ is an element of $\vartheta$

$(2): \quad$ The intersection of any finite number of elements of $\vartheta$ is an element of $\vartheta$.

In Empty Set is Element of Topology it is shown that in any topological space $\left({X, \vartheta}\right)$ it is always the case that $\varnothing \in \vartheta$.

Sigma-Algebra, which looks similar on the surface to a topology, but closed under a countable number of unions. A topology has no such limitation on countability.