Intuitively speaking, an open set is a set without a border: every element of the set has, in its neighborhood, other elements of the set. If, starting from a point of the open set, you move away a little, you never exit the set.
A closed set is the complement of an open set (i.e. what stays "outside" from the open set).
Note that some set exists, that are neither open nor closed.
No, duality and symmetry are not the same thing. Although in many contexts "the dual of" is a symmetric relation, this is not invariably the case (e.g. the dual of the dual of a topological vector space need not be the original).
Moreover symmetry is not just about symmetric relations; it has to do mainly with automorphisms of algebraic, geometric or combinatorial structures. Those structure preserving automorphisms (including trivial the identity mapping) form a group, and we'd refer to it as the symmetry group of the structure.
As you note, there are many kinds of symmetry. Some symmetries have order two but many do not. Indeed the group of symmetries may combine elements that have finite order with those having infinite order, elements that have discrete action with some that are continuous mappings. The symmetries of a right circular cylinder, for example, would include discrete actions like reflection in a midplane as well as continuous actions of rotation about the axis.
If you are looking for a fundamental difference, perhaps it should be noted that duality often involves different categories, i.e. the dual may belong to a different category than the original, while symmetry involves not only the same category but actually a mapping of the same object to itself.
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Let's say you have two expressions $X$ and $Y$. A way of expressing that they are equivalent, or equal to each other by definition, is as follows: $$X\equiv Y.\tag1$$ However, the symbol $\equiv$ also denotes congruence, e.g. $p^2\equiv 1\pmod 6$, so using the symbol for two different circumstances can cause some confusion. Thus, to denote $(1)$, some write that $$X:= Y\quad\text{ or }\quad X\triangleq Y\quad\text{ or }\quad X\stackrel{\text{def}}{=} Y.$$ But, there is also another (but less common) variation, namely, $$X\operatorname*{\equiv}\limits^{\underline{ \ \ \ }}Y\quad\text{ or with a different typeset, }\quad X\,\require{HTML} \style{display: inline-block; transform: rotate(90deg)}{\shortparallel\shortparallel}Y.$$ The "quadruple bar" is not used to denote congruence.
This post might serve useful.