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:

  1. $$\overline{\phi} = \phi$$
  2. $$M \in \mathfrak{P} (S) \implies \overline{M} \supset M$$
  3. $$M, N \in \mathfrak{P}(S) \implies \overline{M} \cup \overline{N} = \overline{M \cup N}$$
  4. $$M \in \mathfrak{P}(S) \implies \overline{\overline{M}} = \overline{M}$$
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:

  1. $$S^○ = S$$
  2. $$M \in \mathfrak{P} (S) \implies M^○ \subset M$$
  3. $$M, N \in \mathfrak{P}(S) \implies M^○ \cap N^○ = (M \cap N)^○$$
  4. $$M \in \mathfrak{P}(S) \implies M^{○○} = M^○$$

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])