= Topological space
{title2=$(X,\tau)$}
{wiki}
A topological space is a <set> $X$ together with a <topology> $\tau\subseteq\mathcal P(X)$, a family of <subsets> satisfying
$$
\varnothing,X\in\tau,\qquad
\bigcup_{i\in I}U_i\in\tau\quad(U_i\in\tau),\qquad
\bigcap_{j=1}^nU_j\in\tau\quad(U_j\in\tau).
$$
Here $\mathcal P(X)$ is the <power set>; the <union> may have arbitrarily many members, while the <intersection> must be finite. The members of $\tau$ are the <open sets>. These are axioms about <subsets>, with no assumed <metric> or surrounding <Euclidean space>. https://www.maths.ox.ac.uk/system/files/attachments/Preliminary%20Reading%20%281%29.pdf[Richard Haydon, Oxford topology preliminaries, Definition 0.1]{ref}
\Image[/finite-topology-examples.png]
{title=Two topologies on the same two-point set and continuity of the identity map}
For $X=\{a,b\}$, the <discrete topology> is $\tau_d=\{\varnothing,\{a\},\{b\},X\}$, while the <indiscrete topology> is $\tau_i=\{\varnothing,X\}$. Both satisfy the axioms. They describe different <topological spaces> on the same <set>: the first is <Hausdorff> and disconnected, while the second is <connected> and not <Hausdorff>. Each is <compact> because $X$ is a <finite set>.
The <identity function> $i:(X,\tau_d)\to(X,\tau_i)$ is a <continuous map>: the only target <open sets> have <inverse images> $\varnothing$ and $X$. Its inverse is not a <continuous map>, since the target <open set> $\{a\}$ has <inverse image> $\{a\}$, which is not an <open set> in $\tau_i$. Thus even a <continuous> <bijection> need not be a <homeomorphism>.
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