Definition:Topological Space

Definition

Let $X$ be any set and let $\vartheta$ be a topology on $X$.

That is, let $\vartheta$ satisfy the axioms:

$(1): \quad$ 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 $\vartheta$

$(3): \quad \varnothing$ and $X$ are both elements of $\vartheta$.

Then $\left({X, \vartheta}\right)$ is called a topological space, or just a space, if the context is clear.

Also, $\left({X, \vartheta}\right)$ can be referred to as the space $X$ if it is clear what topology is actually carried on it.

Note

Note that the properties given here are not strictly as axiomatic as they could be. However, these are the ones which are most often used, as they are in a convenient form.

See the definition of topology to see them pared down to the most basic.

Notation

Some authors use the suboptimal $\left\{{X, \vartheta}\right\}$, which leaves it conceptually unclear as to which is the set and which the topology. This adds unnecessary complexity to the underlying axiomatic justification for the existence of the very object that is being defined.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$