Review of Topological Spaces
Review of Topological Spaces
- Recall from the Topologies and Topological Spaces page that if $E$ is a set then a Topology on $E$ is a collection $\tau$ of subsets of $E$ called Open Sets which satisfy the following properties:
Open Sets of a Topology | |
(1) | $\emptyset$ and $X$ are open sets.. |
(2) | The union of an arbitrary collection of open sets is an open set. |
(3) | The intersection of a finite collection of open sets is an open set. |
- The pair $(E, \tau)$ is called a Topological Space.
- The Closed Sets in a topological space are the complements of open sets. We have the following properties of closed sets:
Closed Sets of a Topology | |
(1) | $\emptyset$ and $X$ are closed sets. |
(2) | The union of an finite collection of closed sets is a closed set. |
(3) | The intersection of an arbitrary collection of closed sets is a closed set. |
- If $E$ is a topological space and $x \in E$ then a Neighbourhood of $x$ is a set $U$ such that there exists an open set $V$ with:
\begin{align} \quad x \in V \subseteq U \end{align}
- We summarize properties of neighbourhoods of points below.
Properties of Neighbourhoods of Points | |
(1) | If $U$ is a neighbourhood of $x$ then $x \in U$. |
(2) | If $U$ and $V$ are neighbourhoods of $x$ then $U \cap V$ is a neighbourhood of $x$. |
(3) | If $U$ is a neighbourhood of $x$ and $U \subseteq V$ then $V$ is a neighbourhood of $x$. |
(4) | If $U$ is a neighbourhood of $x$ then there exists a neighbourhood $V$ of $x$ such that $U$ is a neighbourhood of each $y \in V$. |
- We noted that conversely, if $E$ is a set such that for each $x \in E$ there is a collection of sets $\mathcal U_x$ which satisfy properties (1)-(4) above, then there is a unique topology on $E$ such that for each $x \in E$, $\mathcal U_x$ is a set of neighbourhoods for $x$.
- We said that a topological space $E$ is Hausdorff if for every pair of distinct points $x, y \in E$ there exists open sets $U$ and $V$ with $x \in U$, $y \in V$, and $U \cap V = \emptyset$.
- On The Interior and Closure of a Set of Points page we defined the Interior of $A$, denoted by $\mathrm{int}(A)$, as the set of all Interior Points of $A$, i.e., the set of all $x \in A$ for which there exists an open set $U$ with $x \in U \subseteq A$. We proved that $A$ is open if and only if $A = \mathrm{int}(A)$.
- We defined the Closure of $A$, denoted by $\overline{A}$, as the set of all Points of Closure of $A$, i.e., the set of all $x \in E$ for which $U \cap A \neq \emptyset$ for all neighbourhoods $U$ of $x$. We proved that $A$ is closed if and only if $A = \overline{A}$.
- On the Bases of Neighbourhoods for a Point page we defined a Base of Neighbourhoods of $x$ to be a collection $\mathcal V_x$ of neighbourhoods of $x$ such that for every neighbourhood $U$ of $x$ there is a $V \in \mathcal V_x$ with $x \in V \subseteq U$. We noted that the collection of all open neighbourhoods of $x$ is a base of neighbourhoods of $x$.
- On the The Induced Topology on a Subset of a Topological Space we said that if $A \subseteq E$ then the Induced Topology on $A$ is the topology $\{ A \cap U : U \in \tau \}$. That is, every open set in $A$ is the intersection of $A$ with an open set in $E$. We then observed that every closed set in $A$ is the intersection of $A$ with a closed set in $E$, and every neighbourhood of $x$ in $A$ is the intersection of $A$ with a neighbourhood of $x$ in $E$.
- We summarize some basic results regarding the induced topology.
Properties of the Induced Topology | |
(1) | If $E$ is Hausdorff then $A$ with the induced topology is Hausdorff. |
(2) | If $E$ is first countable then $A$ with the induced topology is first countable. |
(3) | If $E$ is metrizable then $A$ with the induced topology is metrizable. |
- On the Continuous Functions Between Topological Spaces page we said that if $E$ and $F$ are topological spaces then a function $f : E \to F$ is Continuous at $x$ if for every neighbourhood $V_{f(x)}$ of $f(x)$ there exists a neighbourhood $U_x$ of $x$ such that:
\begin{align} \quad f(U_x) \subseteq V_{f(x)} \end{align}
- We say that $f$ is **Continuous if it is continuous at each $x \in E$. We summarize some equivalent continuity criteria below.
Continuity Criteria | |
(1) | $f$ is continuous. |
(2) | For every open set $U$ in $F$, $f^{-1}(U)$ is an open set in $E$. |
(3) | For every closed set $V$ in $F$, $f^{-1}(V)$ is a closed set in $E$. |
- On the Homeomorphisms Between Topological Spaces page we said that a Homeomorphism between $E$ and $F$ is a bijective function $f : E \to F$ for which $f$ and $f^{-1}$ are continuous. We summarize some basic properties of homeomorphisms below.
Properties of Homeomorphisms | |
**(1) | If $f : E \to F$ is a homeomorphism and $A \subseteq E$ then $f(\mathrm{int}(A)) = \mathrm{int}(f(A))$. |
(2) | If $f : E \to F$ is a homeomorphism and $A \subseteq E$ then $f(\overline{A}) = \overline{f(A)}$. |
(3) | $E$ is Hausdorff if and only if $F$ is Hausdorff. |
(4) | $E$ is first countable if and only if $F$ is first countable. |
- On the Metric Spaces and Metrizability page we defined a Metric Space to be a pair $(E, d)$ where $d : E \times E \to [0, \infty)$ has the following properties:
Axioms of a Metric Space | |
(1) | $d(x, y) = 0$ if and only if $x = y$. |
(2) | $d(x, y) = d(y, x)$ for all $x, y \in E$. |
(3) | $d(x, z) \leq d(x, y) + d(y, z)$ for all $x, y, z \in E$. |
- It should be noted that every metric space is a topological space. Furthermore, every metric space is Hausdorff and first countable.