1. Topological Space
Definition. Topology and Topological Space, and Open Sets.
A pair $(X,\Sigma)$ is called a topological space if $X$, as a set, is equipped with a collection $\Sigma$ of subsets (called a topology) of $X$ that satisfies:
-
$\emptyset\in\Sigma$, $X\in\Sigma$
-
Arbitraty union is in $\Sigma$. i.e.
$$\{U_\alpha\}_{\alpha\in\Lambda}\subset\Sigma\implies \bigcup_{\alpha\in\Lambda}U_{\alpha}\in\Sigma $$ -
Finite intersection is in $\Sigma$, i.e.
$$\{U_i\}_{i=1}^n\subset\Sigma\implies \bigcap_{i=1}^nU_{i}\in\Sigma $$
In this case, every $U$ in $\Sigma$ is called an open set.
Example. $(X, d)$ is a metric space,
- $(X, \{\emptyset, X\})$ is called the trivial space.
- $(X, 2^X)$ is called the discrete space.
Definition. Subspace Topology and Subspace
$(X, \Sigma)$ is a topological space with $Y\subset X$. The the collection
is called a subspace topology of $\Sigma$, and $(Y,\Sigma_Y)$ is called a subspace of $(X, \Sigma)$ correspondingly.
Definition. Closedness
$(X, \Sigma)$ is a topological space, $S$ is called closed in $X$ if $X\setminus S\in \Sigma$
Same in metric space, closedness is not contrary to openness.
Lemma. $(X, \Sigma)$ is a topological space, then the followings are true
- $\emptyset$ and $X$ are closed.
- arbitrary intersection of closed subsets are is closed.
- finite union of closed subsets is closed.
Lemma. $Y\subset X$ is a subspace of $X$, then $S$ is closed in $Y$ (with respect to the subspace topology) iff there exists a closed subset $E\subset X$ such that $S=E\cap Y$
Corollary, $Y\subset X$ is closed. $S\subset Y$ is closed in $Y$ iff $S$ is closed in $X$.
Definition. Interior Point Set, Closure
$\Sigma$ is a topology of $X$, the set of Interior Point of $S\subset X$ is defined as
With $S$ defined, the closure of $S$ is defined as
By definition we have
Theorem. Localization Property of Closure