In [[Metrics|metric spaces]], [[Open subsets of metric spaces|open subsets]] (unions of open balls) prescribe which points are 'close' to one another, i.e. for an element $x$ of a metric space $X$, $N\subset X$ is a *neighbourhood* of $x$ iff there is an open ball centred at $x$, $B_{r}(x)$, contained in $N$. The *topology* of the metric space is then defined as the set of all its open subsets. Here, we extend the notion of a topology to any set $X$, that is, we assert that the open subsets are just there, as if given to us from some benevolent power. In fact, the topology may not arise from some metric (if they do then we say the topology is *metrisable*). Topology allows one to express *continuity*, *interior*, *boundary* and *closure* without referring to distance. # Definitions ###### Open set axioms A topology $\mathcal{T}$ of $X$ is a set of [[Subsets|subsets]] of $X$, which we agree to call *open* sets, satisfying the following properties: (T1) $\emptyset, X\in \mathcal{T}$; (T2) finite intersection of open sets are open; (T3) arbitrary unions of open sets are open. The pair $(X, \mathcal{T})$ is called a *topological space*. A set $N\subset X$ is a *neighbourhood* of a point $x\in X$ if and only if there exists an open set $U\in \mathcal{T}$ with $x\in U$ and $U\subset N$. ###### Closed set axioms A subset $S$ of $X$ is a said to be *closed* if and only $X\setminus S$ is open (i.e. $X\setminus S \in \mathcal{T}$). We can assert equivalently that arbitrary intersections and finite unions of closed sets are closed. TBC: Discuss duality. Sets that are both open and closed are sometimes called *clopen* (e.g. $\emptyset$ and $X$). ###### Generating topologies/ topological bases For example, the topology of a metric space is generated by open balls, i.e. any open subset of a metric space is a union of open balls. More generally, given any set $\mathcal{B}$ of subsets of a set $X$, we can generate a topology of $X$ by taking arbitrary unions of elements of $\mathcal{B}$ so long as it satisfies the following axioms: 1. $X= \bigcup_{B\in\mathcal{B}} B$, i.e. $\mathcal{B}$ is a cover of $X$; 2. and for all $B_{1}, B_{2}\in \mathcal{B}$, their intersection $B_{1} \cap B_{2}$ is the union of members of $\mathcal{B}$, i.e. for $x\in B_{1} \cap B_{2}$, there exists $B\in \mathcal{B}$ such that $x\in B$ and $B \subset B_{1} \cap B_{2}$. Such a set $\mathcal{B}$ is known as topological *base* or *basis*. We can prove that the generated set, $\mathcal{T}=\{ \bigcup_{B\in S} B : S \subset \mathcal{B} \}$, is indeed a topology of $X$ (see [[Recognition of topological bases]]). ###### Product topology Let $\mathcal{T}_{X}, \mathcal{T}_{Y}$ be topologies on sets $X,Y$. The [[Product topology|product topology]] on $X\times Y$ is defined as the topology $\mathcal{T}$ generated by the basis $\{ U_{X} \times U_{Y}: U_{X} \in \mathcal{T}_{X}, U_{Y} \in \mathcal{T}_{Y} \}$ . ###### Subspace topology Let $S \subset X$. The *subspace topology* on $S$ is defined by $\mathcal{T}_{S} = \{ U \cap S: U\in \mathcal{T} \}$ (notice that one often denotes an element of $\mathcal{T}$ by $U$). We call $(S,\mathcal{T}_{S})$ a (topological) *subspace* of $(X, \mathcal{T})$. See [[Subspace topology]]. ###### Quotient topology See [[Quotient topology]]. ###### Convergence TBC # Properties ###### Interior, boundary and closure Let $S \subset X$. The [[Interior|interior]], [[Boundary|boundary]] and [[Closure|closure]] of $S$ are defined as follows. The interior of $S$, often denoted $S^{\circ}$ or $\text{int}(S)$, is defined as the union of all open sets contained in $S$. The boundary of $S$, often denoted $\partial S$, is defined as the set of points in $X$ such that every open set containing the point intersects both $S$ and $X \setminus S$ non-trivially. The closure of $S$, often denoted $\overline{S}$, is the defined as the intersection of all closed sets contained in $S$. We shall now list at some of algebraic properties of these operators: 1. Distributivity over unions and intersections: $\overline{H \cup K}= \overline{H} \cup \overline{K}$ and $(H \cap K)^{\circ}=H^{\circ} \cap K^{\circ}$. 2. Complements: $(X\setminus A)^{\circ} =X \setminus \overline{A}$ and $\overline{X\setminus A}= X\setminus A^{\circ}$. 3. Boundary: $\partial A= \overline{A}\cap \overline{X\setminus A}$ and $\partial A=\overline{A}\setminus A^{\circ}$. TBC: draw pictures; find examples for $\overline{H \cap K} \neq \overline{H} \cap \overline{K}$; problem sheet 4. ###### Coarseness TBC: discuss trivial, and discrete topologies. ###### Morphisms ###### Metrisability, Hausdorff property, and other separation axioms See [[Hausdorff topological spaces]] and [[Metrisable topological spaces]]. ###### Compactness See [[Compact topological spaces]]. ###### Connectedness See [[Connected topological spaces]]. # Applications 1. [[Co-finite topology]]. 2. [[Co-countable topology]]. 3. [[Infinitude of primes|Furstenberg's topology]] 4. [[Manifolds]]. 5. [[CW-complexes]].