Topology: An Inquiry-Based Approach

$\mathbb{R}$ has the topology ${\tau }_p$ with $p=0$

$\mathbb{R}$ has the topology ${\tau }_{\bar{p}}$ with $p=0\text{.}$

Show that $\mathcal{B}$ is a basis for a topology ${\tau }_{\ell \ell }$ on $\mathbb{R}\text{.}$ This topology is called the lower limit topology on $\mathbb{R}\text{.}$ The line $\mathbb{R}$ with the topology ${\tau }_{\ell \ell }$ is sometimes called the Sorgenfrey line (after the mathematician Robert Sorgenfrey).

Show that every open interval $(a,b)$ is also an open set in the lower limit topology.

If ${\tau }_1$ and ${\tau }_2$ are topologies on a set $X$ such that ${\tau }_1\subseteq {\tau }_2\text{,}$ then ${\tau }_1$ is said to be a coarser topology that ${\tau }_2\text{,}$ or ${\tau }_2$ is a finer topology that ${\tau }_1\text{.}$ Part (b) shows that the lower limit topology may be a finer topology than the Euclidean metric topology. Determine if this is true, that the lower limit topology is actually a finer topology than the Euclidean metric topology on $\mathbb{R}\text{.}$ Justify your answer.

Let $a<b$ be in $\mathbb{R}\text{.}$ Is the set $[a,b)$ clopen in $(\mathbb{R},{\tau }_{\ell \ell })\text{?}$ Prove your answer.

Show that $\mathbb{Q}$ is dense in $\mathbb{R}$ using the Euclidean metric topology.

Is $\mathbb{Z}$ dense in $\mathbb{R}$ using the Euclidean metric topology? Prove your answer.

Let $A$ be a subset of a topological space $A\text{.}$ Prove that $A$ is dense in $X$ if and only if $A\cap O\ne \mathrm{\varnothing }$ for every open set $O\text{.}$

Show that the sets $\text{Int}(A)\text{,}$ $\text{Bdry}(A)\text{,}$ and $\text{Int}(A^c)$ are mutually disjoint (that is, the intersection of any two of these sets is empty).

Prove that $X=\text{Int}(A)\cup \text{Bdry}(A)\cup \text{Int}(A^c)\text{.}$

$X$ has the topology ${\tau }_p$

$X$ has the topology ${\tau }_{\bar{p}}\text{.}$

If $A$ is a subset of a topological space $X\text{,}$ prove that a point $a\in A$ is an isolated point of $A$ if and only if $\{a\}$ is an open set in $A\text{.}$

We proved that in a metric space every boundary point of a set $A$ is either a limit point or an isolated point of $A\text{.}$ (See Exercise 12.) Is the same statement true in a topological space? Prove your answer.

Show that the collection of sets described as a basis for the Double Origin topology is actually a basis for a topology.

Is $X$ with the Double Origin topology Hausdorff? Prove your answer.

Show that finite sets are closed in ${\mathbb{R}}^n$ with the Zariski topology.

Show that ${\mathbb{R}}^n$ with the Zariski topology is not Hausdorff. (Exercise 12 shows that a basis for the Zariski topology on ${\mathbb{R}}^n$ is the collection of sets of the form ${\mathbb{R}}^n\setminus Z(f)\text{,}$ where $Z(f)$ is the set of zeros of the polynomial $f$ in $n$ variables.)

Prove that every metric space is regular.

Prove that every metric space is a normal space.

Show that $(\mathbb{R},{\tau }_K)$ is a Hausdorff space.

Exercise 16 shows that every metric space is regular. In this part of the exercise, show that $(R,{\tau }_K)$ is not a regular space. We can conclude that $(\mathbb{R},{\tau }_K)$ is not metrizable.

Prove that a topological space $X$ is $T_1$ if and only if each singleton set is closed.

Show that every $T_2$-space is $T_1\text{,}$ that every $T_3$-space is $T_2\text{,}$ and that every $T_4$-space is $T_3\text{.}$

Show that $\mathbb{R}$ with the finite complement topology is $T_1$ but not $T_2\text{.}$

Define the space ${\mathbb{R}}_K$ as in Example 13.13 to be the set of reals with topology $\tau$ with a basis that consists of the standard open intervals in $\mathbb{R}$ along with all sets of the form $(a,b)\setminus K\text{,}$ where $(a,b)$ is any open interval and $K=\{\frac{1}{k}\mid k\in {\mathbb{Z}}^{+}\}\text{.}$ Show that ${\mathbb{R}}_K$ is $T_2$ but not $T_3\text{.}$

Every limit point of a subset $A$ of a topological space $X$ is also a boundary point of $A\text{.}$

Every boundary point of a subset $A$ of a topological space $X$ is also a limit point of $A\text{.}$

If $X$ is a topological space and $A\subseteq X$ such that $\text{Int}(A)=\bar{A}\text{,}$ then $A$ is both open and closed.

If $X$ is a topological space and $A$ and $B$ are subsets of $X$ with $\bar{A}=\bar{B}$ and $\text{Int}(A)=\text{Int}(B)\text{,}$ then $A=B\text{.}$

If $A$ and $B$ are subsets of a topological space $X\text{,}$ then $\overline{A\cap B}=\bar{A}\cap \bar{B}\text{.}$

If $A$ and $B$ are subsets of a topological space $X\text{,}$ then $\overline{A\cup B}=\bar{A}\cup \bar{B}\text{.}$