Topological Spaces

Let $X$ be a set. A topology on $X$ is a collection $\mathcal{T}$ of subsets of $X$ that satisfies the topology axioms:

• $\varnothing\in\mathcal{T}$ and $X\in\mathcal{T};$
• the union of any collection of sets in $\mathcal{T}$ is in $\mathcal{T};$
• the intersection of finitely many sets in $\mathcal{T}$ is in $\mathcal{T}.$

Let $X$ be a set and $\mathcal{T}$ be a topology on $X.$ The pair $\left(X,\mathcal{T}\right)$ is a topological space, and is written $X$ when the topology is understood. The sets in $\mathcal{T}$ are the open sets of the topological space.

Let $\left(X,\mathcal{T}\right)$ be a topological space and $F\subseteq X.$ The set $F$ is closed if and only if its complement $X\setminus F$ is open. Arbitrary intersections and finite unions of closed sets are closed.

Let $(X,d)$ be a metric space. The metric topology induced by $d$ is the collection of open sets of the metric space, which satisfies the topology axioms. Every metric space is a topological space with its metric topology, and equivalent metrics induce the same topology.

Let $X$ be a set. The discrete topology on $X$ is the topology in which every subset of $X$ is open, and is induced by the discrete metric. The indiscrete topology on $X$ is the topology $\left\{\varnothing,X\right\}.$

Let $\left(X,\mathcal{T}\right)$ be a topological space. The topological space is metrisable if and only if its topology is induced by some metric. The indiscrete topology on a set with more than one point is not metrisable.

Let $X$ be a set and $\mathcal{T}$ and $\mathcal{T}'$ be topologies on $X.$ The topology $\mathcal{T}'$ is finer than $\mathcal{T},$ and $\mathcal{T}$ is coarser than $\mathcal{T}',$ if and only if $\mathcal{T}\subseteq\mathcal{T}'.$ The discrete topology is the finest topology on $X$ and the indiscrete topology is the coarsest.

Let $\left(X,\mathcal{T}\right)$ be a topological space and $x\in X.$ A neighbourhood of $x$ is a set that contains an open set containing $x.$

Let $\left(X,\mathcal{T}\right)$ be a topological space and $A\subseteq X.$ The interior points, limit points and isolated points of $A,$ and the interior, closure and boundary of $A$ and the density of $A$ in $X,$ are defined as in a metric space with “open ball centred at $x$” replaced by “neighbourhood of $x$”. The interior remains the largest open set contained in $A$ and the closure remains the smallest closed set containing $A.$

Let $\left(X,\mathcal{T}\right)$ be a topological space, $\left(x_n\right)$ be a sequence in $X$ and $x\in X.$ The sequence $\left(x_n\right)$ converges to $x$ if and only if every neighbourhood of $x$ contains all but finitely many of its terms.

Let $\left(X,\mathcal{T}\right)$ be a topological space. A basis for $\mathcal{T}$ is a collection $\mathcal{B}\subseteq\mathcal{T}$ of open sets such that every open set is a union of sets in $\mathcal{B}.$ The open balls form a basis for the metric topology, as do the open balls with rational radii, and the open intervals with rational endpoints form a countable basis for the usual topology on $\mathbb{R}.$

Let $X$ be a set and $\mathcal{B}$ be a collection of subsets of $X.$ The collection $\mathcal{B}$ is a basis for some topology on $X$ if and only if $\mathcal{B}$ covers $X$ and the intersection of any two sets in $\mathcal{B}$ is a union of sets in $\mathcal{B}.$ The topology generated by $\mathcal{B}$ is the coarsest topology containing $\mathcal{B}.$

Let $\left(X,\mathcal{T}\right)$ be a topological space and $A\subseteq X.$ The subspace topology on $A$ is the collection of intersections of $A$ with the open sets of $X.$

$$\mathcal{T}_A:=\left\{U\cap A:U\in\mathcal{T}\right\}$$

Let $\left(X,\mathcal{T}\right)$ be a topological space and $A\subseteq X.$ The set $A$ with the subspace topology is a subspace of $X.$ Subspaces have the following properties:

• a metric subspace carries the subspace topology;
• a set that is open in a subspace need not be open in the whole space: $\left[0,\frac{1}{2}\right)$ is open in the subspace $\left[0,1\right]$ of $\mathbb{R};$
• if $A$ is open in $X$ then the open sets of the subspace are exactly the open subsets of $X$ contained in $A,$ and likewise for closed sets if $A$ is closed.

Let $\left(X,\mathcal{T}_X\right)$ and $\left(Y,\mathcal{T}_Y\right)$ be topological spaces. The product topology on the Cartesian product $X\times Y$ is the topology generated by the basis of products of open sets.

$$\mathcal{B}:=\left\{U\times V:U\in\mathcal{T}_X,\,V\in\mathcal{T}_Y\right\}$$

The product of finitely many spaces is defined in the same way. The product topology has the following properties:

• the product topology on $\mathbb{R}^m\times\mathbb{R}^n$ is the usual topology on $\mathbb{R}^{m+n};$
• the product of the metric spaces $\left(X,d_X\right)$ and $\left(Y,d_Y\right)$ is metrisable by the product metric $d\left(\left(x,y\right),\left(x',y'\right)\right):=\max\left\{d_X\left(x,x'\right),d_Y\left(y,y'\right)\right\},$ which is strongly equivalent to $d_X+d_Y$ and to $\sqrt{d_X^2+d_Y^2};$
• a sequence in a product converges if and only if each of its coordinate sequences converges.

Let $\left(X,\mathcal{T}_X\right)$ and $\left(Y,\mathcal{T}_Y\right)$ be topological spaces and $f:X\rightarrow Y$ be a function. The function $f$ is continuous if and only if the inverse image of every open set is open.

$$f^{-1}(V)\in\mathcal{T}_X\,\,\forall\,V\in\mathcal{T}_Y$$

For metric spaces this agrees with the earlier definition. Equivalently, $f$ is continuous if and only if the inverse image of every closed set is closed, or if and only if the inverse image of every set in a basis for $\mathcal{T}_Y$ is open.

Let $X$ and $Y$ be topological spaces. The coordinate projections of the product $X\times Y$ are the functions $\left(x,y\right)\mapsto x$ and $\left(x,y\right)\mapsto y.$ Continuous functions between topological spaces have the following properties:

• constant functions, the identity function, the inclusion of a subspace and the coordinate projections of a product are continuous;
• a composition of continuous functions is continuous;
• the restriction of a continuous function to a subspace is continuous;
• a function into a product is continuous if and only if each of its coordinate functions is continuous;
• a continuous function maps convergent sequences to convergent sequences, but in a general topological space the converse can fail, so continuity is not determined by sequences alone.

Let $X$ and $Y$ be topological spaces and $f:X\rightarrow Y$ be a function. The function $f$ is a homeomorphism if and only if $f$ is a bijection and both $f$ and $f^{-1}$ are continuous. The spaces $X$ and $Y$ are homeomorphic if and only if there exists a homeomorphism between them. A homeomorphism induces a bijection between the topologies of $X$ and $Y.$

A topological property is a property of topological spaces that is preserved by homeomorphisms. Being metrisable and having a countable basis are topological properties. Boundedness and completeness are not topological properties, since $x\mapsto\tan x$ is a homeomorphism from the bounded incomplete space $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ onto $\mathbb{R}.$

Let $X$ and $Y$ be topological spaces and $f:X\rightarrow Y$ be a continuous function. The function $f$ is an open map if and only if it maps open sets to open sets. The function $f$ is a closed map if and only if it maps closed sets to closed sets. A continuous bijection need not be a homeomorphism: $t\mapsto e^{it}$ is a continuous bijection from $\left[0,2\pi\right)$ onto the unit circle whose inverse is discontinuous at $1.$ A continuous bijection is a homeomorphism if and only if it is an open map, if and only if it is a closed map.

Let $\left(X,\mathcal{T}\right)$ be a topological space. The space $X$ is Hausdorff if and only if any two distinct points have disjoint neighbourhoods.

$$\forall\,x\neq y\in X\,\exists\,U,V\in\mathcal{T}:x\in U,\,y\in V,\,U\cap V=\varnothing$$

Hausdorff spaces have the following properties:

• every metric space is Hausdorff, since $B\left(x,\frac{r}{2}\right)$ and $B\left(y,\frac{r}{2}\right)$ are disjoint when $r=d(x,y);$
• the indiscrete topology on a set with more than one point is not Hausdorff;
• subspaces and products of Hausdorff spaces are Hausdorff;
• in a Hausdorff space finite sets are closed and a convergent sequence has exactly one limit, whereas in the indiscrete topology every sequence converges to every point.