Constructions:: *[[Constructions|Used in the construction of...]]* Specializations:: *[[Specializations]]* Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* Properties:: *[[Properties]]* Examples:: [[discrete topology]], [[discrete topology|indiscrete topology]], [[finite complement topology]], [[topology generated by a basis]], [[standard topology on the real line]], [[lower limit topology, upper limit topology]], [[order topology]], [[Zariski Topology of varieties]] Nonexamples:: *[[Nonexamples]]* ---- > [!definition] Definition. ([[topological space]]) > A **topology** on a set $X$ is a collection $\tau=\tau_{X}$ of subsets of $X$, called **the open sets in $X$**, having the properties > 1. $\emptyset, X \in \tau$; > 2. (Arbitrary unions of open sets are open) The union of the elements of any subcollection of $\tau$ is again in $\tau$; >3. (Finite intersections of open sets are open) The intersection of the elements of any finite subcollection of $\tau$ is again in $\tau$. > > A **topological space** is a pair $(X,\tau)$, where $X$ is a set and $\tau$ a **topology** on $X$. > > Topological spaces are elements of [[category]] $\mathsf{Top}$. The homset $\text{Hom}(X,Y)$ is the set of [[continuous]] maps $X \to Y$. ^3fd880 > [!intuition] > Intuition comes from the topology of $\mathbb{R}^{n}$ that one uses in real analysis before taking a point-set topology course. There, if we specialize our definition of open sets of $\mathbb{R}^{n}$ to [[open sets can be nestled in|be those for which an open ball can nestle around every point]] (the definition used in real analysis of $\mathbb{R}^{n}$) > [!basicnonexample] > The set $\{ U \subset \mathbb{R} : U \text{ is infinite} \} \cup \{ \emptyset \}$ is *not* a topology on $\mathbb{R}$, for it is not closed under finite intersection: $[0,1] \cap [1,2]= \{ 1 \}$ is not infinite nor empty. ^a512b0 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```