MA260-Norms-Metrics-and-Topologies-Revision

My own notes about the MA260 Norms, Metrics and Topologies revision, mainly from the notes and example sheets.

MA260-Norms-Metrics-and-Topologies-Revision
- Lecture notes
- Example sheets

Lecture notes

260_notes_2022-23.pdf

First of all, feel free to download 2022-2023 lecture notes for MA260.

Example sheets

MA260_Norms_metrics_topologies_tutorial_sheet.pdf

For the second term revision, I will combine all the examples into one pdf, and therefore we will not use sub-headings for each week but instead just topics.

Countable

A set $A$ is countable if it is in bijection with a subset of $\mathbb{N}$ or, equivalently, if there is an injection from $A$ to $\mathbb{N}$. A countable set may be finite or infinite, in which case we call it countably infinite.
The empty set is countable.

Norms

A norm on a vector space $X$ is a map $\left|\cdot\right|:X\to\mathbb{R}^{+}$ such that

$\left|x\right| = 0$ if and only if $x = 0$.
$\left|\lambda x\right| = \left|\lambda\right|\left|x\right|$ for every $\lambda\in\mathbb{R}$ or $\mathbb{C}$, $x\in X$ ('homogeneity') and
$\left|x+y\right|\leq\left|x\right|+\left|y\right|$ for every $x,y\in X$ (the triangle inequality).

Standard norm/Euclidean norm

In the vector space $\mathbb{R}^{n}$, for $x = (x_{1},x_{2},...,x_{n})$ define $$\left|x\right| = \left(\sum_{j=1}^{n}\left|x_{j}\right|^{2}\right)^{1/2},$$ the standard norm or Euclidean norm.
There are other norms on $\mathbb{R}$. For example, $$\left|x\right|_ {l^{1}} = \sum_{j=1}^{n}\left|x_{j}\right|$$ and $$\left|x\right|_ {l^{\infty}} = \max_{j=1,...,n}\left|x_{j}\right|.$$

Normed space

If $X$ is a vector space and $\left|\cdot\right|$ is a norm on $X$, the pair $(X,\left|\cdot\right|)$ is a normed space.

Closed unit ball

The closed unit ball in $(X,\left|\cdot\right|)$ is the set $$\mathcal{B}_{X} = \set{x\in X: \left|x\right|\leq 1}.$$

Convex

Let $X$ be a vector space. A subet $K$ of $X$ is convex if whenever $x,y\in K$ and $0\leq\lambda\leq 1$, we have $\lambda x+ (1-\lambda)y\in K$.

Minkowski's inequality in $\mathbb{R}^{n}$

For all $1\leq p\leq\infty$, if $x,y\in\mathbb{R}^{n}$, then $$\left|x+y\right|_ {l^{p}}\leq\left|x\right|_ {l^{p}} + \left|y\right|_ {l^{p}}.$$

Equivalent norms

Two norms $\left|\cdot\right|_ {1}$ and $\left|\cdot\right|_ {2}$ on $X$ are equivalent if there exists constants $0 \leq c_{1}\leq c_{2}$ such that $$c_{1} \left|x\right|_ {1}\leq\left|x\right|_ {2}\leq c_{2}\left|x\right|_ {1},$$ for every $x\in X$.

$l^{p}$ space

The $l^{p}$ norm is given by $$\left|x\right|_ {l^{p}} = \left(\sum_{j=1}^{\infty}\left|x_{j}\right|^{p}\right)^{1/p}.$$

Minkowski's inequality in $l^{p}$

For all $1\leq p\leq\infty$, if $x,y\in l^{p}$ then $x+y\in l^{p}$ and $$\left|x+y\right|_ {l^{p}}\leq\left|x\right|_ {l^{p}} + \left|y\right|_ {l^{p}}.$$

$L^{p}$ space

For $p\in [1,\infty)$, set $$\left|f\right|_ {L^{p}} = \left(\int_{a}^{b}\left|f(x)\right|^{p}\mathrm{d}x\right)^{1/p}.$$

Metric space

A metric $d$ on a set $X$ is a map $d:X\times X\to\mathbb{R}^{+}$ such that

$d(x,y) = 0$ if and only if $x = y$.
$d(x,y) = d(y,x)$ for every $x,y\in X$ and
$d(x,z)\leq d(x,y)+d(y,z)$ for every $x,y,z\in X$ (triangle inequality).

Standard metric/Euclidean metric

The standard metric or Euclidean metric on $\mathbb{R}^{n}$ is given by $$d_{2}(x,y) = \left|x-y\right|_ {l^{2}} = \left(\sum_{j=1}^{n}\left|x_{j}-y_{j}\right|^{2}\right)^{1/2}.$$

Discrete metric

The Discrete metric on any non-empty set $X$ is defined by setting $d(x,x) = 0$ and $d(x,y) = 1$ if $x\ne y$. (Useful in counterexamples)

Sunflower metric

The Sunflower metric on $\mathbb{R}^{2}$ is given by $d(x,y) = \left|x-y\right|$ if $x$ and $y$ lie on same line through the origin, and $\left|x\right|+\left|y\right|$ otherwise.

Jungle river metric

The Jungle river metric on $\mathbb{R}^{2}$ is given by $d((x_{1},y_{1}),(x_{2},y_{2})) = \left|y_{1}-y_{2}\right|$ if $x_{1}= x_{2}$ and $d((x_{1},y_{1}),(x_{2},y_{2})) = \left|y_{1}\right|+\left|x_{1}-x_{2}\right|+\left|y_{2}\right|$ otherwise.

Open and closed sets

The Open ball centred at $a\in X$ of radius $r$ is the set $$\mathbb{B}(a,r) = \set{x\in X: d(x,a) < r}.$$
The Closed ball centred at $a\in X$ of radius $r$ is the set $$\overline{\mathbb{B}}(a,r) = \set{x\in X: d(x,a)\leq r}.$$

Bounded subsets

A subset $S$ of $(X,d)$ is bounded if there exists $a\in X$ and $r > 0$ such that $S\subset\mathbb{B}(a,r)$.

Open and closed subsets

A subset $U$ of $(X,d)$ is open in $X$ if for every $x\in U$, there exists $\varepsilon > 0$ such that $\mathbb{B}(x,\varepsilon)\subset U$.
A subset $F$ of $(X,d)$ is closed in $X$ if $X\setminus F$ is open.
Open balls are open.
If $U_{1},...,U_{n}$ are open in $(X,d)$, then $\displaystyle\bigcap_{i=1}^{n} U_{i}$ is open in $(X,d)$.
If $F_{1},...,F_{n}$ are closed in $(X,d)$, then $\displaystyle\bigcup_{i=1}^{n} F_{i}$ is closed in $(X,d)$.
If $\set{U_{i}:i\in\mathcal{I}}$ is any collection of sets that are open in $(X,d)$, where $\mathcal{I}$ is any index set, then $U = \displaystyle\bigcup_{i\in\mathcal{I}} U_{i}$ is open in $(X,d)$.
If $\set{F_{i}: i\in\mathcal{I}}$ is any collection of closed sets in $(X,d)$ then $\displaystyle\bigcap_{i\in\mathcal{I}} F_{i}$ is closed in $(X,d)$.

Convergence of sequences

Convergence

A sequence $(x_{n})_ {n=1}^{\infty}$ in $(X,d)$ converges to $x\in X$ if $$\lim_{n\to\infty} d(x_{n},x) = 0,$$ in terms of open balls this can be phrased as for every $\varepsilon > 0$ there exists $N\geq 1$ such that $$x_{n}\in\mathbb{B}(x,\varepsilon)$$ for all $n\geq N$.
A subset $F$ of a metric space is closed if and only if whenever a sequence $(x_{n})_{n=1}^{\infty}$ contained in $F$ converges to some $x\in X$, it follows that $x\in F$.

Continuity

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces and let $f:X\to Y$ be a function. For $p\in X$, we say that $\displaystyle\lim_{x\to p}f(x) = y\in Y$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $$0 < d_{X}(X, p) < \delta\implies d_{Y}(f(x),y) < \varepsilon. $$
Let $(X, d_{X})$ and $(Y, d_{Y})$ be metric spaces and let $f: X\to Y$ be a function. Then $f$ is continuous at $p\in X$ if $\displaystyle\lim_{x\to p}f(x) = f(p)$, i.e. if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that $$d_{X}(x, p) < \delta\implies d_{Y}(f(x), f(p)) < \varepsilon.$$
It is continuous on $X$ if it is continuous at every point of $X$.
A function $f:X\to Y$ is Lipschitz continuous or just Lipschitz if there exists $C\geq 0$ such that $$d_{Y}(f(x),f(y))\leq Cd_{X}(x,y)$$ for every $x,y\in X$.
Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. A function $f:X\to Y$ is continuous if and only if for any open set $U\subset Y, f^{-1}(U)$ is open in $X$.

Continuity and open/close sets

If $f:\mathbb{R}^{n}\to\mathbb{R}^{k}$ is continuous at all points of $\mathbb{R}^{n}$, then for all open subsets $V$ of $\mathbb{R}^{k}$, $f^{-1}(V)$ is open; for all closed subsets $\mathcal{F}$ of $\mathbb{R}^{k}$, $f^{-1}(\mathcal{F})$ is closed.

Topologically equivalence

Suppose that $d_{1}$ and $d_{2}$ are two metrics on $X$. Then the following statements are equivalent:

every set that is open in $(X,d_{2})$ is open in $(X,d_{1})$;
for any metric space $(Y,d_{Y})$, if $g:X\to Y$ is continuous from $(X,d_{2})$ into $(Y,d_{Y})$, then $g$ is continuous from $(X,d_{1})$ into $(Y,d_{Y})$ and
for any metric space $(Y,d_{Y})$, if $f:Y\to X$ is continuous from $(Y,d_{Y})$ into $(X,d_{1})$ then $f$ is continuous from $(Y,d_{Y})$ into $(X,d_{2})$.

Suppose that $d_{1}$ and $d_{2}$ are two metrics on $X$. Then the following statements are equivalent:

the open sets in $(X,d_{1})$ and $(X,d_{2})$ coincide;
for any metric space $(Y,d_{Y})$, a function $g:X\to Y$ is continuous from $(X,d_{1})$ into $(Y,d_{Y})$ if and only if $g$ is continuous from $(X,d_{2})$ into $(Y,d_{Y})$;
for any metric space $(Y,d_{Y})$, a function $f:Y\to X$ is continuous from $(Y,d_{Y})$ into $(X,d_{1})$ if and only if $f$ is continuous from $(Y,d_{Y})$ into $(X,d_{2})$.

Topologically equivalent

Two metrics $d_{1}$ and $d_{2}$ on $X$ are called topologically equivalent or just equivalent if the open setes in $(X,d_{1})$ and $(X,d_{2})$ coincide.
Two metrics $d_{1}$ and $d_{2}$ on $X$ are called Lipschitz equivalent if there exists $0 < c\leq C < \infty$ such that $$cd_{1}(x,y)\leq d_{2}(x,y)\leq Cd_{1}(x,y)$$ for all $x,y\in X$.

Equivalent norms

The metrics induced by equivalent norms are topologically equivalent.
If $X$ is a vector space and two norms $\left|\cdot\right|_ {1}$ and $\left|\cdot\right|_ {2}$ on $X$ induce topologically equivalent metrics then the norms are equivalent.

Isometries and homeomorphisms

Suppose that $f:X\to Y$ is a bijection such that $$d_{Y}(f(x),f(y)) = d_{X}(x,y)$$ for all $x,y\in X$. Then $f$ is called an isometry between $X$ and $Y$. It preserves the distance between points, so $X$ and $Y$ are the same as metric spaces. We say that $X$ and $Y$ are isometric.
If $f:X\to Y$ is a bijection and both $f$ and $f^{-1}$ are continuous, we say that $f$ is a homeomorphism and that $X$ and $Y$ are homeomorphic.

Topological properties I

If some property $P$ of a metric space is such that if $(X,d)$ has property $P$, then so does every metric space that is homeomorphic to $(X,d)$ we say that $P$ is a topological property.

Examples of topological properties:

$X$ is open in $X$; $X$ is closed in $X$.
$X$ is finite; countably infinite or uncountable.
Every continuous real-valued function on $X$ is bounded.

Examples of properties that are not topological:

$X$ is bounded.
$X$ is totally bounded: for each $r > 0$ there exists a finite set $F$ such that every ball of radius $r$ contains a point of $F$.

Topological Spaces

Definition of a topology

A topology $\mathcal{T}$ on a set $T$ is a collection of subsets of $T$, which we agree to call the "open sets", such that

$T$ and $\emptyset$ are open;
the intersection of finitely many open sets is open; and
arbitrary unions of open sets are open.

The pair $(T,\mathcal{T})$ is called a topological space.

Definition of finer and coarser

If $\mathcal{T}_ {1}$ and $\mathcal{T}_ {2}$ are two topologies on $T$ then we say that $\mathcal{T}_ {1}$ is coarser than $\mathcal{T}_ {2}$ if $\mathcal{T}_ {1}\subset \mathcal{T}_ {2}$, i.e. $\mathcal{T}_ {1}$ contains "fewer" open sets than $\mathcal{T}_ {2}$. In this situation, we also say that $\mathcal{T}_ {2}$ is finer than $\mathcal{T}_ {1}$.

Closed topological space

A subset of a topological space $T$ is closed if its complement is open. Using De Morgan's laws, the collection $\mathcal{F}$ of all closed sets satisfies

$T$ and $\emptyset$ are closed.
the union of finitely many closed sets is closed; and
arbitrary intersections of closed sets are closed.

Bases and sub-bases

A basis for a topology $\mathcal{T}$ on $T$ is a collection $\mathcal{B}\subset\mathcal{T}$ such that every set in $\mathcal{T}$ is the union of some sets from $\mathcal{B}$, i.e. for all $U\in\mathcal{T}$, there exists $\mathcal{C}_ {U}\subset\mathcal{B}$ such that $U = \displaystyle\bigcup_{B\in\mathcal{C} _ {U}}B$.

If $\mathcal{B}$ is any basis for $\mathcal{T}$, then

$T$ is the union of some sets from $\mathcal{B}$ (i.e. there exists $\mathcal{C}_ {T}\subset\mathcal{B}$ such that $\displaystyle\bigcup_{B\in\mathcal{C}_ {T}} B = T)$;
If $B_{1},B_{2}\in\mathcal{B}$, then $B_{1}\cap B_{2}$ is the union of some sets from $\mathcal{B}$ (i.e. there exists $\mathcal{C}_ {B_{1}\cap B_{2}}\subset\mathcal{B}$ such that $\displaystyle\bigcup_{B\in\mathcal{C}_ {B_{1}\cap B_{2}}}B = B_{1}\cap B_{2}$.
A sub-basis for a topology $\mathcal{T}$ on $T$ is a collection $\mathcal{B}\subset\mathcal{T}$ such that every set in $\mathcal{T}$ is a union of finite intersections of sets from $\mathcal{B}$.

Subspaces and finite product spaces

Subspace topology

If $(T,\mathcal{T})$ is a topological space and $S\subset T$, then the subspace topology on $S$ is $$\mathcal{T}_ {S} = \set{U\cap S:U\in\mathcal{T}}.$$ We call $(S,\mathcal{T}_ {S})$ a topological subspace of $T$.

Product topological spaces

Suppose that $(T_{1},\mathcal{T}_ {1})$ and $(T_{2}, \mathcal{T}_ {2})$ are two topological spaces. Then the product topology on $T_{1}\times T_{2}$ is the topology $\mathcal{T}$ with basis $$\mathcal{B} = \set{U_{1}\times U_{2}: U_{1}\in\mathcal{T}_ {1}, U_{2}\in\mathcal{T}_ {2}}.$$ We call $(T_{1}\times T_{2}, \mathcal{T})$ the topological product of $T_{1}$ and $T_{2}$.

Closure, interior and boundary

A neighbourhood of $x\in T$ is a set $H\subset T$ such that $x\in U\subset H$ for some $U\in\mathcal{T}$. An open neighbourhood of $x\in T$ is an open set $U$ that contains $x$.

Closure

The closure $\overline{A}$ of a set $A\subset T$ is the intersection of all closed sets that contain $A$.
The closure of $A$, $\overline{A}$ is the set $$\overline{A} = \set{x\in T:U\cap A\ne\emptyset\text{for every open set U that contains x}} = \set{x\in T:\text{every neighbourhood of x intersects A}}.$$
If $X$ is a metric space and $A\subset X$ then $$\overline{A} = \set{\text{limits of convergent sequences in A}}.$$

Interior

The interior of $A$, $A^{\circ}$, is the union of all open subsets of $A$.
The interior of $A$, $A^{\circ}$ consists of all points for which $A$ is a neighbourhood, i.e. $$\set{x\in T:x\in U\subset A, \text{for some} U\in\mathcal{T}}.$$

Relationship between closure and interior

If $A\subset T$, then $$A^{\circ} = T\setminus\overline{T\setminus A}\quad\text{and}\quad\overline{A} = T\setminus(T\setminus A)^{\circ}.$$

Boundary

The boundary $\partial H$ of a set $H$ is the set of all points $x$ whith the property that every neighbourhood of $x$ meets both $H$ and its complement: $$\partial H = \set{x\in T: \text{if U is an open set that contains x then} U\cap H\ne\emptyset, U\cap(T\setminus H)\ne\emptyset}.$$
We have $$\partial H = \overline{H}\setminus H^{\circ}.$$

Limit point and Isolated point

Let $S\subset T$. A point $x\in T$ is a limit point of $S$ if every neighbourhood of $x$ intersects $S\setminus\set{x}$.
A point in $S$ that is not a limit point of $S$ is called an isolated point.

Dense, nowhere dense and meagre

A subset $A$ of $T$ is

dense in $T$ if $\overline{A} = T$;
nowhere dense in $T$ if $(\overline{A})^{\circ} = \emptyset$;
meagre in $T$ if it is a union of a countable number of nowhere dense sets.

The Hausdorff property and metrisability

A topological space $(T,\mathcal{T})$ is metrisable if there is a metric $d$ on $T$ such that $\mathcal{T}$ consists of the open sets in $(T,d)$.
A sequence $(x_{n})^{\infty}_ {n=1}$ in a topological space $T$ converges to $x$ if for every open neighbourhood $U$ of $x$ there exists $N\geq 1$ such that $x_{n}\in U$ for all $n\geq N$.
A topological space $T$ is Hausdorff if for any two distinct $x,y\in T$, there exist disjoint open sets $U,V$ such that $x\in U$ and $y\in V$.
In a Hausdorff space $T$, any sequence has at most one limit.

Continuity between topological spaces

A map $f:T_{1}\to T_{2}$ between two topological spaces $(T,\mathcal{T}_ {1})$ and $(T_{2},\mathcal{T}_ {2})$ is continuous if whenever $U\subset T_{2}$ is open, $f^{-1}(U)$ is open in $T_{1}$. (i.e. if $U\in\mathcal{T}_ {2}$, then $f^{-1}(U)\in\mathcal{T}_ {1}$).
Suppose that $f:T_{1}\to T_{2}$ is a map between two topological spaces $(T_{1},\mathcal{T}_ {1})$ and $(T_{2},\mathcal{T}_ {2})$, and that $\mathcal{B}$ is a sub-basis for the topology $\mathcal{T}_ {2}$. Then $f$ is continuous if and only if $f^{-1}(B)$ is open in $T_{1}$ for every $B\in\mathcal{B}$.

Basic properties

Transitivity of continuity

If $(T_{1}, \mathcal{T}_ {1})$, $(T_{2},\mathcal{T}_ {2})$ and $(T_{3}, \mathcal{T}_ {3})$ are topological spaces and $f: T_{1}\to T_{2}$ and $g: T_{2}\to T_{3}$ are continuous, then $g\circ f: T_{1}\to T_{3}$ is continuous.

Projection continuity

For $j = 1,2$, the projection $\pi_{j}: T_{1}\times T_{2}\to T_{j}$ is continuous.
Let $(T,\mathcal{T}), (T_{1},\mathcal{T}_ {1})$ and $(T_{2},\mathcal{T}_ {2})$ be topological spaces. A map $f = (f_{1},f_{2}):T\to T_{1}\times T_{2}$ is continuous if and only if $f_{1}$ and $f_{2}$ are both continuous. (i.e. $\pi_{1}\circ f$ and $\pi_{2}\circ f$ are continuous).

Additivity, product and division continuity

If $f,g:T\to\mathbb{R}$ are continuous, then so are $f+g, fg$ and $f/g$ is continuous on the set $\set{x\in T: g(x)\ne 0}$.

The projective topology and product spaces

The projective topology on $T$ is the coarsest topology for which all the maps $f_{j}:T\to T_{j}$ are continuous.
Let $(T_{j},\mathcal{T}_ {j}), j\in J$ be an arbitrary collection of topological spaces. Their product $T = \displaystyle\prod_{j\in J} T_{j}$ is the set of all functions $x:J\to\displaystyle\bigcup_{j\in J} T_{j}$ such that $x(j)\in T_{j}$. The product topology $\mathcal{T}$ on $T$ is the coarsest topology for which all of the projections $$\pi_{j}: T\to T_{j}:x\mapsto x(j)$$ are continuous. We then call the topological space $(T,\mathcal{T})$ the topological product of the spaces $(T_{j},\mathcal{T}_ {j})$.
A sub-basis for the product topology consists of all sets of the form $$\prod_{j\in J} U_{j},$$ where $U_{j}\in\mathcal{T}_ {j}$ with $U_{j} = T_{j}$ except for a finite number of the $j$.

Homeomorphisms

Let $(T_{1},\mathcal{T}_ {1})$ and $(T_{2}, \mathcal{T}_ {2})$ be topological spaces. A bijection $f: T_{1}\to T_{2}$ is a homeomorhpism if any one of the following equivalent conditions holds:

both $f$ and $f^{-1}$ are continuous;
$V$ is open in $T_{2}$ if and only if $f^{-1}(V)$ is open in $T_{1}$.
$U$ is open in $T_{1}$ if and only if $f(U)$ is open in $T_{2}$.

If there is a homemorphism $f:T_{1}\to T_{2}$, we say that $(T_{1},\mathcal{T}_ {1})$ and $(T_{2},\mathcal{T}_ {2})$ are homeomorphic.

A property of topological spaces is a topological invariant (or 'topological property') if it is preserved by homeomorphisms.

Compactness

Cover and subcover

A cover of a set $A$ is collection $\mathcal{U}$ of sets whose union contains $A$: $$A\subset\bigcup_{U\in\mathcal{U}} U.$$
A subcover of a cover $\mathcal{U}$ is a subset of $\mathcal{U}$ whose elements still cover $A$.
A cover is open if all of its elements are open.

Compact

A topological space $T$ is compact if every open cover of $T$ has a finite subcover.
A subset $S$ of $T$ is compact if every open cover of $S$ by subsets of $T$ has a finite subcover. This is the same as $S$ being compact with the subspace topology.
If $T$ is a topological space and $S\subset T$, then $S$ is compact in the sense of definition of compact if and only if $(S,\mathcal{T}_ {S})$ is compact in the sense of definition of compact.

Heine-Borel Theorem

Any closed interval $[a,b]$ is a compact subset of $\mathbb{R}$ (with the usual topology).

Compact vs closed

Any closed subset $S$ of a compact space $T$ is compact.
Any compact subset $K$ of a Hausdorff space $T$ is closed.
Any compact subset $K$ of a metric space $(X,d)$ is bounded.
A subset of $\mathbb{R}$ with the usual topology is compact if and only if it is closed and bounded.
Let $\mathcal{F}$ be a collection of non-empty closed subsets of a compact space $T$ such that every finite subcollection of $\mathcal{F}$ has a non-empty intersection. Then the intersection of all the sets from $\mathcal{F}$ is non-empty.
Let $F_{1}\supset F_{2}\supset F_{3}\supset...$ be non-empty closed subsets of a compact space $T$. Then $\displaystyle\bigcap_{j=1}^{\infty} F_{j}\ne\emptyset.$

Compactness of products and compact subsets of $\mathbb{R}^{n}$

If $T$ and $S$ are compact topological spaces, then $T\times S$ is compact.
The product of a finite numnber of compact spaces is compact.

Tychonov's Theorem

The product of any collection of compact spaces is compact (with the product topology).

Heine-Borel in $\mathbb{R}^{n}$

A subset of $\mathbb{R}^{n}$ is compact if and only if it is closed and bounded.

Continuous functions on compact sets

A continuous image of a compact space is compact.
A continuous bijection of a compact space $T$ onto a Hausdorff space $S$ is a homeomorphism.

Semicontinuous

A function $f:T\to\mathbb{R}$ is lower semicontinuous if for every $c\in\mathbb{R}$, the set $f^{-1}(c,\infty)$ is open.
It is upper semicontinuous if for every $c\in\mathbb{R}$, the set $f^{-1}(-\infty.c)$ is open.
If $f:T\to\mathbb{R}$ is both upper and lower semicontinuous, then $$f^{-1}(a,\infty)\cap f^{-1}(-\infty,b) = f^{-1}(a,b)$$ is open for every $a,b\in\mathbb{R}$, and so $f$ is continuous.
If $T$ is non-empty and compact and $f:T\to\mathbb{R}$ is lower semicontinuous then it is bounded below and attains its minimum.
If $f$ is upper semicontinuous, then it is bounded above and attains its maximum.
If $T$ is non-empty and compact, then a continuous function $f:T\to\mathbb{R}$ is bounded and attains its bounds.

Equivalence of all norms on $\mathbb{R}^{n}$

All norms on $\mathbb{R}^{n}$ are equivalent.

Lebesgue numbers and uniform continuity

Let $\mathcal{U}$ be an open cover of a metric space $(X,d)$. A number $\delta > 0$ is called a Lebesgue number for $\mathcal{U}$ if for any $x\in X$, there exists $U\in\mathcal{U}$ such that $\mathbb{B}(x,\delta)\subset U$.
Every open cover $\mathcal{U}$ of a compact metric space $(X,d)$ has a Lebesgue number.

Uniformly continuous

A map $f:(X,d_{X})\to (Y, d_{Y}) is uniformly continuous if for every $\varepsilon > 0$, there exists $\delta > 0$ such that $$d_{X}(x,y) < \delta \implies d_{Y}(f(x),f(y)) < \varepsilon$$ for any $x,y\in X$.
A continuous map from a compact metric space into a metric space is uniformly continuous.

Sequential compactness

A subset $K$ of a metric space $(X,d)$ is sequentially compact if every sequence in $K$ has a convergent subsequence whose limit lies in $K$.
If $K$ is a sequentially compact subset of a metric space, then any open cover of $K$ has a Lebesgue number.
A subset of a metric space is sequentially compact if and only if it is compact.

Normed spaces

A normed space is finite-dimensional if and only if its closed unit ball is compact.

Connectedness

We say that a pair of sets $(A,B)$ is a partition of a topological space $T$ if $T = A\cup B$ and $A\cap B = \emptyset$; and we then say that $A$ and $B$ partition $T$.

Connected/Disconnected

A topological space $T$ is connected if the only partitions of $T$ into open sets are $(T,\emptyset)$ and $(\emptyset, T)$. The space $T$ is said to be disconected if it is not connected.

The following are equivalent:

$T$ is disconnected;
$T$ has a partition into two non-empty open sets.
$T$ has a partition into two non-empty closed sets.
$T$ has a subset that is both open and closed and is neither $\emptyset$ nor $T$.
There is a continuous function from $T$ onto the two-point set $\set{0,1}$ with the discrete topology.

Subset connected/disconnected

A subset $S$ of $T$ is connected/disconnected if $(S,\mathcal{T}_ {S})$ is connected/disconnected.
A set $S\subset T$ is separated by subsets $U,V\in\mathcal{T}$ if $$S\subset U\cup V,\quad U\cap V\cap S = \emptyset,\quad U\cap S\ne\emptyset,\quad V\cap S\ne\emptyset.$$
A subspace $S$ of a topological space $T$ is disconnected if and only if it is separated by some open subsets $U,V\in\mathcal{T}.$

Connected subsets of $\mathbb{R}$

A set $I\subset\mathbb{R}$ is an interval if and only if whenever $x,y\in I$ and $x < z < y$, we have $z\in I$.
A subset of $\mathbb{R}$ is connected if and only if it is an interval.

Operations on connected sets

Suppose that $C_{j}, j\in\mathcal{J}$, are connected subsets of $T$ and $C_{i}\cap C_{j}\ne\emptyset$ for each $i,j$, then $$K = \bigcup_{j\in\mathcal{J}} C_{j}$$ is connected.
Suppose that $C_{1}$ and $C_{2}$ are connected subsets of $T$ and $\overline{C}_ {1}\cap C_{2}\ne\emptyset$. Then $C_{1}\cup C_{2}$ is connected.
If $C\subset T$ is connected, then so is any set $K$ satisfying $C\subset K\subset\overline{C}$.
The continuous image of a connected set is connected.
The product of two connected spaces is connected.
If $T$ and $S$ are homeomorphic and $T$ is connected, then $S$ is connected.

Topologist sine curve

The 'topologist's sine curve' is given by $$\mathcal{F} = \set{\left(x,\sin\frac{1}{x}\right):x\in\mathbb{R},x\ne 0}\cup\set{(0,0)}$$ and it is connected.

Connected components

The equivalence class of $\sim$ are called the connected components of $T$.

Path-connected spaces

If $u,v\in T$, a path from $u$ to $v$ is a continuous map $\varphi:[0,1]\to T$ such that $\varphi(0) = u$ and $\varphi(1) = v$.
A space $T$ is path connected if any two points in $T$ can be joined by a path in $T$.
A path-connected space $T$ is connected.

Open sets in $\mathbb{R}^{n}$

Connected open subsets of $\mathbb{R}^{n}$ are path connected.
Open subsets of $\mathbb{R}^{n}$ have open connected components.
A subset $U$ of $\mathbb{R}$ is open if and only if it is the disjoint union of countably many open intervals, i.e. $U = \displaystyle\bigcup_{j\in\mathcal{J}}(a_{j}, b_{j})$, with the intervals disjoint and $\mathcal{J}$ finite or countably infinite.

Cauchy sequence

If a sequence $(x_{n})$ converges in a metric space $(X,d)$, then it is Cauchy, i.e. for every $\varepsilon > 0$, there exists $N$ such that $$d(x_{n},x_{m}) < \varepsilon$$ for every $n,m\geq N$.

Completeness

A metric space $(X,d)$ is complete if any Cauchy sequence in $X$ converges.
Suppose that $(X,d)$ is a metric space and that $S$ is a subset of $X$. If $(S,d|_ {S})$ is complete, then $S$ is a closed subset of $X$, and if $(X,d)$ is complete and $S$ is closed, then $(S,d|_ {S})$ is complete.
Any compact metric space $(X,d)$ is complete.

Examples of complete spaces

$\mathbb{R}^{d}$ is complete.
For every $1\leq p\leq\infty$, $l^{p}$ is complete.
For any non-empty set $X$, the space $B(X)$ of bounded real-valued functions on $X$, $f:X\to\mathbb{R}$, with the 'sup norm' $$\left|f\right|_ {\infty} = \sup_{x\in X}\left|f(x)\right|$$ is complete.
The space $C_{b}(T)$ of all bounded continuous functions from any non-empty topological space $T$ into $\mathbb{R}$ is a closed subspace of $B(T)$, and hence complete.
If $T$ is non-empty and compact, then $C(T)$ is complete with the maximum norm $$\left|f\right|_ {\infty} = \max_{x\in T}\left|f(x)\right|.$$

Completions

Any metric space $(X,d)$ can be isometrically embedded into the complete metric space $B(X)$.
Any metric space has a completion.