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Some of the non-Hausdorff topologies that turn up are actually not that hard to get an intuition for. For example, you can think of the Zariski topology on a classical algebraic variety $V$ as just being a collection of information describing all the closed subvarieties of $V$ (e.g., the Zariski topology on $\mathbb{A}^3_k$ describes all the algebraic curves, surfaces, and points in 3-space).

It might seem at first glance that the topologies involved get hard to understand when we move from varieties to schemes, but really the topology of a scheme is not hard to get a handle on either. The key to understanding the topologies of schemes is to understand the generic points and to understand these you just need to get some intuition about the concept of specialization and generalization.

Given two points $x,y$ in a topological space $X$, we say that $x$ is a specialization of $y$ (or that $y$ is a generalization of $x$) if $x$ is contained in the closure of $y$. What this means is that $y$ is contained in every open neighbourhood of $x$. I like to think of this as meaning that $y$ is infinitesimally close to $x$. Similarly, given a subset $F\subseteq X$ we say that a point $x\in F$ is a generic point of $F$ if $F$ is the closure of $x$. Evidently a necessary condition for such an $F$ to possess a generic point is that $F$ be a (non-empty) irreducible closed subset of $X$. It is not hard to show that in a $T_0$-space every irreducible closed subset has at most one generic point. But in fact the topology of a scheme is nicer than this: the topology of a scheme has the nice property that every (non-empty) irreducible closed subset has a unique generic point. (Such as space is called a sober space).

How should we think about this? Well, if $F$ is a closed irreducible subset of $X$ and $\xi$ is a generic point of $F$ then this means that every point of $F$ is a specialization of $\xi$; in other words $\xi$ is contained in every open neighbourhood of every point in $F$. So this generic point is infinitesimally close to all of the points in $F$. Now, in a sober space the map sending a point to its closure provides a bijection between the set of points of the space and the set of non-empty irreducible closed subsets of the space. So if you take any scheme $X$, the closed points are the points that you should think of as being the points forming a "geometric space", and all the other points are simply generic points of the various irreducible closed subsets of this space--each non-closed point describes a unique irreducible closed subset.

For example, consider the scheme version of the affine plane: $\mathbb{A}^2_k=Spec(k[X,Y])$. The subspace of closed points (i.e., the maximal ideals) is homeomorphic to the usual variety affine plane with the Zariski topology; all the other points of the scheme are just generic points describing all the subvarieties of the affine plane.

Some of this may be a bit vague or imprecise, but the point is that it isn't too hard to develop some intuition for the (non-Hausdorff) topologies arising in algebraic geometry.

Some of the non-Hausdorff topologies that turn up are actually not that hard to get an intuition for. For example, you can think of the Zariski topology on a classical algebraic variety $V$ as just being a collection of information describing all the closed subvarieties of $V$ (e.g., the Zariski topology on $\mathbb{A}^3_k$ describes all the algebraic curves, surfaces, and points in 3-space).
Given two points $x,y$ in a topological space $X$, we say that $x$ is a specialization of $y$ (or that $y$ is a generalization of $x$) if $x$ is contained in the closure of $y$. What this means is that $y$ is contained in every open neighbourhood of $x$. I like to think of this as meaning that $y$ is infinitesimally close to $x$. Similarly, given a subset $F\subseteq X$ we say that a point $x\in F$ is a generic point of $F$ if $F$ is the closure of $x$. Evidently a necessary condition for such an $F$ to possess a generic point is that $F$ be a (non-empty) irreducible closed subset of $X$. It is not hard to show that in a $T_0$-space every irreducible closed subset has at most one generic point. But in fact the topology of a scheme is nicer than this: the topology of a scheme has the nice property that every (non-empty) irreducible closed subset has a unique generic point. (Such as space is called a sober space).
How should we think about this? Well, if $F$ is a closed irreducible subset of $X$ and $\xi$ is a generic point of $F$ then this means that every point of $F$ is a specialization of $\xi$; in other words $\xi$ is contained in every open neighbourhood of every point in $F$. So this generic point is infinitesimally close to all of the points in $F$. Now, in a sober space the map sending a point to its closure provides a bijection between the set of points of the space and the set of non-empty irreducible closed subsets of the space. So if you take any scheme $X$, the closed points are the points that you should think of as being the points forming a "geometric space", and all the other points are simply generic points of the various irreducible closed subsets of this space--each non-closed point describes a unique irreducible closed subset.
For example, consider the scheme version of the affine plane: $\mathbb{A}^2_k=Spec(k[X,Y])$. The subspace of closed points (i.e., the maximal ideals) is homeomorphic to the usual variety affine plane with the Zariski topology; all the other points of the scheme are just generic points describing all the subvarieties of the affine plane.