Topological Spaces: General Properties

Show that the discrete topology for a non-empty set $X$ is a metric topology.

Show that the trivial topology on a set has a unique base.

Regarding Proposition 2, show that if $\mathcal{B}$ is a base for a topology, then properties (i) and (ii) hold.

Proposition 2: For a non-empty set $X$, let $\mathcal{B}$ be a collection of subsets of $X$. Then $\mathcal{B}$ is a base for a topology for $X$ if and only if (i) $\mathcal{B}$ covers $X$, that is, $X = \bigcup_{B \in \mathcal{B}} B$; and (ii) if $B_{1}$ and $B_{2}$ are in $\mathcal{B}$ and $x \in B_{1} \cap B_{2}$, then there is a set $B$ in $\mathcal{B}$ for which $x \in B \subseteq B_{1} \cap B_{2}$.

Let $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ be topologies for a non-empty set $X$. Show that $\mathcal{T}_{1} = \mathcal{T}_{2}$ if and only if there are bases $\mathcal{B}_{1}$ for $\mathcal{T}_{1}$ and $\mathcal{B}_{2}$ for $\mathcal{T}_{2}$ that are related as follows at each point $x$ in $X$ : for each neighborhood $\mathcal{N}_{1}$ of $x$ belonging to $\mathcal{B}_{1}$, there is a neighborhood $\mathcal{N}_{2}$ of $x$ belonging to $\mathcal{B}_{2}$ for which $\mathcal{N}_{2} \subseteq \mathcal{N}_{1}$ and for each neighborhood $\mathcal{N}_{2}$ of $x$ belonging to $\mathcal{B}_{2}$, there is a neighborhood $\mathcal{N}_{1}$ of $x$ belonging to $\mathcal{B}_{1}$ for which $\mathcal{N}_{1} \subseteq \mathcal{N}_{2}$.

Let $E$ be a subset of a topological space $X$.

Let $\mathcal{O}$ be an open subset of a topological space $X$. For a subset $E$ of $X$, show that $\mathcal{O}$ is disjoint from $E$ if and only if it is disjoint from $\bar{E}$.

Show that the collection of intervals of the form $[a, b)$, where $a < b$, is a base for a topology for the set of real numbers $\mathbb {R}$. The set of real numbers $\mathbb {R}$ with this topology is called the Sorgenfrey Line.

Consider the upper half-plane, $\mathbb {R}^{2,+} = \left\{ (x, y) \in \mathbb {R}^{2} \mid y \geq 0\right\}$. For points $(x, y)$ with $y > 0$, take as a basic open neighborhood a usual Euclidean open ball centered at ($x, y$) and contained in the upper half-plane. As a basic open neighborhood of a point $(x, 0)$ take the set consisting of the point itself and all the points in an open Euclidean ball in the upper half-plane that is tangent to the real line at ($x, 0$). Show that this collection of sets is a base. The set $\mathbb {R}^{2,+}$ with this topology is called the Moore Plane.

Show that if $F$ is a closed subset of a normal space $X$, then the subspace $F$ is normal. Is it necessary to assume that $F$ is closed?

Let $X$ be a topological space. Show that $X$ is Hausdorff if and only if the diagonal $D = \left\{ \left(x_{1}, x_{2}\right) \in X \times X \mid x_{1} = x_{2}\right\}$ is a closed subset of $X \times X$.

Consider the set of real numbers with the topology that has as a subbase and sets of the form $(-\infty , c), c \in \mathbb {R}$. Show that this space is not Tychonoff.

In $\mathbb {R}^{n}$ let $\mathcal{B}$ be the family of sets $\left\{ x \in \mathbb {R}^{n} \mid p(x) \neq 0\right\}$, where $p$ is a polynomial in $n$ variables. Let $\mathcal{T}$ be the topology on $X$ that has $\mathcal{B}$ as a subbase. Show that $\mathcal{T}$ is a topology for $\mathbb {R}^{n}$ that is Tychonoff but not Hausdorff.

Show the Sorgenfrey Line and the Moore Plane are Hausdorff (see Problems 7 and 8).

A topological space is said to be a Lindelöf space or to have the Lindelöf property provided that each open cover of $X$ has a countable subcover. Show that if $X$ is second countable, then it is Lindelöf.

Let $X$ be an uncountable set of points, and let $\mathcal{T}$ consist of $\emptyset$ and all subsets of $X$ that have finite complements. Show that $\mathcal{T}$ is a topology for $X$ and that the space ($X, \mathcal{T}$) is not first countable.

Show that a second countable space is separable and every subspace of a second countable space is second countable.

Show that the Moore Plane is separable (see Problem 8). Show that the subspace $\mathbb {R} \times \left\{ 0\right\}$ of the Moore Plane is not separable. Conclude that the Moore Plane is not metrizable and not second countable.

Show that the Sorgenfrey Line is first countable but not second countable and yet the rationals are dense (see Problem 7). Conclude that the Sorgenfrey Line is not metrizable.

Let $X_{1} = N \times N$, where $N$ denotes the set of natural numbers and take $X = X_{1} \cup \left\{ \omega \right\}$, where $\omega$ does not belong to $X_{1}$. For each sequence $s = \left\{ m_{k}\right\}$ of natural numbers and natural number $n$, define

Let $f$ be a mapping of the topological space $X$ to the topological space $Y$ and $\mathcal{S}$ be a subbase for the topology on $Y$. Show that $f$ is continuous if and only if the inverse image under $f$ of every set in $\mathcal{S}$ is open in $X$.

Let $X$ be a topological space.

For topological spaces $X$ and $Y$, let $f$ map $X$ to $Y$. Which of the following assertions are equivalent to the continuity of $f$ ? Verify your answers.

Prove Proposition 11.

Proposition 11: The composition of continuous mappings between topological spaces is continuous.

Prove Proposition 12.

Proposition 12: Let $X$ be a non-empty set and $\mathcal{F} = \left\{ f_{\lambda }: X \rightarrow X_{\lambda }\right\}_{\lambda \in \Lambda }$ be a collection of mappings parametrized by a set $\Lambda$, where each $X_{\lambda }$ is a topological space. The topology $\mathcal{T}$ for $X$ that has the collection of sets $\mathcal{C} = \left\{ f_{\lambda }^{-1}(O_{\lambda }) \mid \lambda \in \Lambda , O_{\lambda } \text{ open in } X_{\lambda }\right\}$ as a subbase is the weakest among the topologies on $X$ for which each mapping $f_{\lambda }: X \rightarrow X_{\lambda }$ is continuous. We call $\mathcal{T}$ the weak topology induced by $\mathcal{F}$.

Prove that the sum and product of two real-valued continuous functions defined on a topological space $X$ are themselves continuous.

Let $\mathcal{F}$ be a collection of real-valued functions on a set $X$. Find necessary and sufficient conditions on $\mathcal{F}$ in order that $X$, considered as a topological space with the weak topology induced by $\mathcal{F}$, is Tychonoff.

For topological spaces $X$ and $Y$, let the mapping $f: X \rightarrow Y$ be one-to-one and onto. Show that the following assertions are equivalent.

For topological spaces $X$ and $Y$, let $f$ be a continuous mapping from $X$ onto $Y$. If $X$ is Hausdorff, is $Y$ Hausdorff? If $X$ is normal, is $Y$ normal?

Show that the inverse of a homeomorphism is a homeomorphism and the composition of two homeomorphisms, when defined, is again a homeomorphism.

Suppose that a topological space $X$ has the property that every continuous real-valued function on $X$ takes a minimum value. Show that any topological space that is homeomorphic to $X$ also possesses this property.

Suppose that a topological space $X$ has the property that every continuous real-valued function on $X$ has an interval as its image. Show that any topological space that is homeomorphic to $X$ also possesses this property.

Show that $\mathbb {R}$ is homeomorphic to the open bounded interval (0,1), but is not homeomorphic to the closed bounded interval $[0,1]$.

Let $X$ and $Y$ be topological spaces and consider a mapping $f$ from $X$ to $Y$. Suppose $X = X_{1} \cup X_{2}$ and the restrictions of $f$ to the topological subspaces $X_{1}$ and to $X_{2}$ are continuous. Show that $f$ need not be continuous at any point in $X$. Show that $f$ is continuous on $X$ if $X_{1}$ and $X_{2}$ are open. Compare this with the case of measurable functions and the inheritance of measurability from the measurability of restrictions.

Show that for any two numbers $a$ and $b$,

A topological space is said to be countably compact provided that every countable open cover has a finite subcover. For a second countable space $X$, show that $X$ is compact if and only if it is countably compact.

Let $X$ be a topological space. Prove that $X$ is countably compact if and only if whenever $\left\{ F_{n}\right\}$ is a descending sequence of non-empty closed subsets of $X$, the intersection $\bigcap_{n = 1}^{\infty } F_{n}$ is non-empty.

Let $X$ be compact, Hausdorff and $\left\{ F_{n}\right\}_{n = 1}^{\infty }$ be a descending collection of closed subsets of $X$. Let $\mathcal{O}$ be a neighborhood of the intersection $\bigcap_{n = 1}^{\infty } F_{n}$. Show there is an index $N$ such that $F_{n} \subseteq \mathcal{O}$ for $n \geq N$.

Show that it is not possible to express a closed, bounded interval of real numbers as the pairwise disjoint union of a countable collection (having more than one member) of closed, bounded intervals.

Let $f$ be a continuous mapping of the compact space $X$ onto the Hausdorff space $Y$. Show that any mapping $g$ of $Y$ into $Z$ for which $g \circ f$ is continuous must itself be continuous.

Let ($X, \mathcal{T}$) be a topological space.

Let $X$ and $Y$ be Hausdorff topological spaces and $Y^{X}$ the collection of maps from $X$ into $Y$. On $Y^{X}$ we define a topology, called the compact-open topology, by taking as a subbase sets of the form $\mathcal{U}_{K, \mathcal{O}} = \left\{ f: X \rightarrow Y \mid f(K) \subseteq \mathcal{O}\right\}$, where $K$ is a compact subset of $X$ and $\mathcal{O}$ is an open subset of $Y$. Thus the compact-open topology is the weakest topology on $Y^{X}$ such that the sets $\mathcal{U}_{K, \mathcal{O}}$ are open.

Let $\left\{ f_{n}\right\}$ be a sequence of continuous real-valued functions on a countably compact space $X$. Suppose that for each $x \in X$, the sequence $\left\{ f_{n}(x)\right\}$ decreases monotonically to zero. Show that $\left\{ f_{n}\right\}$ converges to zero uniformly.

Let $\left\{ C_{\lambda }\right\}_{\lambda \in \Lambda }$ be a collection of connected subsets of a topological space $X$ and suppose that any two of them have a point in common. Show that the union of $\left\{ C_{\lambda }\right\}_{\lambda \in \Lambda }$ also is connected.

Let $A$ be a connected subset of a topological space $X$, and suppose that $A \subseteq B \subseteq \bar{A}$. Show that $B$ is connected.

Show that the following subset of the plane is connected but not arcwise connected.

Show that an arcwise connected topological space $X$ is connected. Also show that each connected open subset $\mathcal{O}$ of a Euclidean space $\mathbb {R}^{n}$ is arcwise connected.

Consider the circle $C = \left\{ (x, y) \mid x^{2}+y^{2} = 1\right\}$ in the plane $\mathbb {R}^{2}$. Show that $C$ is connected.

Show that $\mathbb {R}^{n}$ is connected.

Show that a compact metric space ($X, \rho$) fails to be connected if and only if there are two disjoint, non-empty subsets $A$ and $B$ whose union is $X$ and $\epsilon > 0$ such that $\rho (u, v) \geq \epsilon$ for all $u \in A, v \in B$. Show that this is not necessarily the case for non-compact metric spaces.

Show that for any point ($x, y$) in the plane $\mathbb {R}^{2}$, the subspace $\mathbb {R}^{2} \sim \left\{ (x, y)\right\}$ is connected. Use this to show that $\mathbb {R}$ is not homeomorphic to $\mathbb {R}^{2}$.

Verify the equivalence of the three assertions in (4).