Special Point-sets #

Special points and sets #

Suppose $(X, G)$ is a topological space. And $S \subseteq X$.

• $x \in S$ is an interior point of $S$ in $X$,
  • iff $x \in O$ for some $O \in G$ and $O \subseteq S$;
  • iff $S \in N _ G(x)$;
• $S^\circ$ is the interior of $S$ in $X$,
  • iff $S^\circ$ is the set of all interior points of $S$,
  • iff $S^\circ$ is the union of all open sets of $X$ that is a subset of $S$;
  • iff $S^\circ$ is the largest open subset of $X$ that is a subset of $S$;
• $x \in X$ is an adherent point of $S$ in $X$,
  • iff every neighborhood of $x$ contains a point of $S$.
• $\overline S$ is the closure $S$ in $X$,
  • iff $\overline S$ is the set of all adherent points of $S$ in $X$;
  • iff $\overline S$ is the intersection of all closed set $F \supseteq S$;
  • iff $\overline S$ is the smallest closed subset of $X$ that contains $S$.
  • $\overline S$ is closed in $X$.
• $x \in X$ is an accumulation point of $S$ in $X$,
  • iff every neighborhood of $x$ contains a point of $S- \c x$.
  • If $x$ is an accumulation point of $S$, it is an adherent point of $S$.
• $S'$ is the derived set of $S$ in $X$,
  • iff $S'$ is the set of all accumulation points of $S$ in $X$.
• $S$ is closed in $X$,
  • iff $S$ contains all its adherent points in $X$;
  • iff $S$ contains all its accumulation points in $X$;
  • iff $S$ equals to its closure in $X$;
• $x \in X$ is an isolated point of $S$ in $X$ iff $x \in S - S'$.
• $x\in X$ is an boundary point of $S$ in $X$ iff $x \in \overline S \cap \overline{S^c}$.
• $\part S = \overline S - S^\circ$ is the boundary of $S$ in $X$.
  • $\part S$ is closed in $X$.

Disjoint sets #

Suppose $(X, G)$ is a topological space. And $A, A _ k, B$ are subsets of $X$.

• If $A^\circ \cap B^\circ = \varnothing$, $A$ and $B$ are almost disjoint.
• If $A \cap B = \varnothing$, $A$ and $B$ are disjoint sets.

Topological subspace #

Suppose $(X, G)$ is a topological space. And $Y \subseteq X$.

The following are equivalent and gives a topological subspace $Y$.

• Modify the neighborhood topology $N'(y) = \{B \cap Y \mid B \in N(y)\}$.
• Modify the set of open sets $G'= \{U \cap Y \mid U \in G\}$.

Consider the topological subspace $(Y, G')$. Suppose $S \subseteq Y$.

• $S$ is open in $Y$ iff $S = A \cap Y$ for some open set $A$ in $X$.
• $S$ is closed in $Y$ iff $S = A \cap Y$ for some closed set $A$ in $X$.

Dense sets #

Suppose $X$ is a topological space. And $E \subseteq X$.

• When $\overline E = X$, $E$ is called dense in $X$.
  • $E$ is dense if and only if every nonempty open subset of $X$ contains a element of $E$.
  • $E$ is dense if and only if $X \backslash E$ has no interior points.
• When $(\overline E)^\circ = \varnothing$ in $E$, $E$ is called nowhere dense in $X$.
• When $\overline E$ has no interior points, $E$ is called nowhere dense.

Topological Properties #

Separable #

Topological space $X$ is separable if it has a countable dense subset.

First-countable #

A topological space $X$ is first-countable if for any $x \in X$ there exists a sequence $N _ 1, N _ 2, \ldots \in N(x)$ such that for any $O \in N(x)$ there is an $N _ k$ contained in $O$.

Without loss of generality, we can require the sequence to contain only open neighbors.

Second-countable #

A topological space $X$ is second-countable if there exists some countable collection $U =\left\{U _ {i}\right\} _ {i=1}^{\infty}$ of open subsets of $X$ such that any open subset of $X$ can be written as a union of elements of some subfamily of $U$.

• Suppose $X$ has a metric, then separable is equivalent to second-countable.
• The Euclidean spaces are second-countable.

Hausdorff space #

Suppose $X$ is a topological space.

If for all $x, y \in X$ where $x \neq y$, there exists $U \in N(x)$ and $V \in N(y)$, where $U \cap V = \varnothing$. $X$ is called Hausdorff.

Compactness of topological spaces #

Suppose $X$ is a topological space with open sets topology $G$.

The space $X$ is called compact if all of its open covers have a finite subcover.

Consider subset $K \subseteq X$. $K$ is called compact, if the subspace $(K, G| _ K)$ is compact.

Compact subspaces #

Suppose $X$ is a compact topological space.

Suppose $F$ is closed in $X$, then $F$ is a compact subspace.

• Suppose $\p{G _ \alpha} _ {\alpha \in I}$ is an open cover of $F$ in $F$.
• Consider $\p{G _ \alpha \cup \p{X - F}}$, which is an open cover of $X$.
• Therefore there is a finite open subcover.

Suppose $X$ is Hausdorff. Suppose $F$ is a compact subspace of $X$. Then $F$ is closed in $X$.

• Suppose $F$ is not closed in $X$, and $p \in \overline F - F$.
• Since $X$ is Hausdorff, there exists open neighborhoods $O _ x \in N(x)$ and $P _ x \in N(p)$. Such that $O _ x \cap P _ x = \varnothing$.
• Consider the open covering $O:=\c{O _ x: x \in F}$.
• Now there exists some open subcover $O'$ of $F$.
• Therefore $p \notin \overline F$. Contradiction!