Topology.Operator (topology v0.1.5)
Operators on topological spaces.
lemma
Lemma
Let $(S, \mathfrak{O})$ be a topological space and (M \subset S) be a subset of (S) . Then the following statements are equivalent:
$$ \overline{M^c} = (M^{○})^c $$
Link to this section Summary
Functions
Returns a closure of the given set for topological space.
Returns a interior of the given set for topological space.
Link to this section Functions
Link to this function
closure_operator(m, arg)
@spec closure_operator( MapSet.t(), {Topology.underlying_set(), Topology.topology()} ) :: MapSet.t() | {:error, <<_::368>>}
Returns a closure of the given set for topological space.
mathematical-expression
Mathematical expression
$$ \overline{M},\ M^c $$
theorem
Theorem
Closure operator has the following properties:
- $$\overline{\phi} = \phi$$
- $$M \in \mathfrak{P} (S) \implies \overline{M} \supset M$$
- $$M, N \in \mathfrak{P}(S) \implies \overline{M} \cup \overline{N} = \overline{M \cup N}$$
- $$M \in \mathfrak{P}(S) \implies \overline{\overline{M}} = \overline{M}$$
Link to this function
interior_operator(m, arg)
@spec interior_operator( MapSet.t(), {Topology.underlying_set(), Topology.topology()} ) :: MapSet.t() | {:error, <<_::368>>}
Returns a interior of the given set for topological space.
mathematical-expression
Mathematical expression
$$ M^○,\ M^i $$
theorem
Theorem
Interior operator has the following properties:
- $$S^○ = S$$
- $$M \in \mathfrak{P} (S) \implies M^○ \subset M$$
- $$M, N \in \mathfrak{P}(S) \implies M^○ \cap N^○ = (M \cap N)^○$$
- $$M \in \mathfrak{P}(S) \implies M^{○○} = M^○$$
examples
Examples
iex> topological_space =
...> {
...> MapSet.new([:a, :b, :c]),
...> MapSet.new([
...> MapSet.new([]),
...> MapSet.new([:b]),
...> MapSet.new([:a, :b]),
...> MapSet.new([:a, :b, :c])
...> ])
...> }
iex> m = MapSet.new([:a, :b])
iex> Topology.Operator.interior_operator(m, topological_space)
MapSet.new([:a, :b])