Summary

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Section Summary

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Important ideas that we discussed in this section include the following.

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A subset $C$ of a topological space $X$ is closed if $X ∖ C$ is open.
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Any intersection of closed sets is closed, while unions of finitely many closed sets are closed.
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A sequence in a topological space $X$ is a function $f : Z^{+}$ to $X .$
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A sequence $(x_{n})$ in a topological space $X$ converges to a point $x$ in $X$ if for each open set $O$ containing $x,$ there exists a positive integer $N$ such that $x_{n} \in O$ for all $n \geq N .$
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If a sequence $(x_{n})$ in a topological space $X$ converges to a point $x,$ then $x$ is a limit of the sequence $(x_{n}) .$
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A limit point of a subset $A$ of a topological space $X$ is a point $x \in X$ such that every neighborhood of $x$ contains a point in $A$ different from $x .$ A subset $C$ of a topological space $X$ is closed if and only if $C$ contains all of its limit points.
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A boundary point of a subset $A$ of a topological space $X$ is a point $x \in X$ such that every neighborhood of $x$ contains a point in $A$ and a point in $X ∖ A .$ The boundary of $A$ is the set

$Bdry (A) = {x \in X ∣ x is a boundary point of A} .$

A subset $C$ of $X$ is closed if and only if $C$ contains its boundary.
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A topological space $X$ is Hausdorff if we can separate distinct points with open sets in the space. That is, if for each pair $x,$ $y$ of distinct points in $X,$ there exists open sets $O_{x}$ of $x$ and $O_{y}$ of $y$ such that $O_{x} \cap O_{y} = \emptyset .$ Hausdorff spaces are important because sequences have unique limits in Hausdorff spaces and single point sets are closed.
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Separation axioms tell us what kinds of objects can be separated by open sets.
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  In a $T_{1}$ -space, we can separate two distinct points with one open set. That is, given distinct points $x$ and $y$ in a $T_{1}$ topological space $X,$ there is an open set $U$ that separates $y$ from $x$ in the sense that $y \in U$ but $x \notin U .$
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  In a $T_{2}$ -space $X$ we can separate points more distinctly. That is, if $x$ and $y$ are different points in $X,$ there exist disjoint open sets $U$ and $V$ such that $x \in U$ and $y \in V .$
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  In a $T_{3}$ -space $X$ we can separate a point from a closed set that does not contain that point. That is, if $C$ is a closed subset of $X$ and $x$ is a point not in $C,$ there exists disjoint open sets $U$ and $V$ in $X$ such that $C \subseteq U$ and $x \in V .$
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  In a $T_{4}$ -space $X$ we can separate disjoint closed sets. That is, if $C$ and $D$ are disjoint closed subsets of $X,$ there exist disjoint open sets $U$ and $V$ such that $C \subseteq U$ and $D \subseteq V .$

Topology: An Inquiry-Based Approach

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Section Summary