topology (point-set topology, point-free topology)
see also algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
A function $f \colon X \to Y$ is called continuous if its values $f(x)$ do not “jump” with variation of its argument $x$, unless $x$ itself “jumps”. Roughly speaking, if $x_1 \approx x_2$, then $f(x_1) \approx f(x_2)$. (This can be made into a precise definition in nonstandard analysis if care is taken about the domains of these variables.)
In order to make this precise (in standard analysis) one needs some concept of neighbourhoods of elements of $X$ and $Y$.
For instance if $X$ and $Y$ carry structure of metric spaces, then one may say that $f$ is continuous if for every point $x \in X$ and for every small open ball around its image $f(x)$ in $Y$, there exists a sufficiently small open ball around $x \in X$ which is still mapped by $f$ into that target open ball. This definition turns out to have more elegant formulation that needs to mention neither the points of $x$ nor the radii of open balls around points: the metric induces a concept of open subsets and $f$ is continuous precisely if preimages under $f$ of open subsets in $Y$ are still open subsets in $X$.
This then is the general definition of continuity of a function $f$ between topological spaces:
A function between topological spaces is continuous precisely if its preimages of open subsets are again open subsets.
Continuous maps are the homomorphisms between topological spaces. In other words, the collection of topological spaces forms a category, often denoted Top, whose morphisms are the continuous functions.
Further generalization of the concept of continuity exists, for instance to locales (and then to toposes) or to convergence spaces. (See also at continuous space.)
We state the definition of continuity in terms of epsilontic analysis, definition 3 below. First recall the relevant concepts:
A metric space is
a set $X$ (the “underlying set”);
a function $d \;\colon\; X \times X \to [0,\infty)$ (the “distance function”) from the Cartesian product of the set with itself to the non-negative real numbers
such that for all $x,y,z \in X$:
$d(x,y) = 0 \;\Leftrightarrow\; x = y$
(symmetry) $d(x,y) = d(y,x)$
(triangle inequality) $d(x,y)+ d(y,z) \geq d(x,z)$.
Every normed vector space $(V, {\Vert -\Vert})$ becomes a metric space according to def. 1 by setting
Let $(X,d)$, be a metric space. Then for every element $x \in X$ and every $\epsilon \in \mathbb{R}_+$ a positive real number, write
(epsilontic definition of continuity)
For $(X,d_X)$ and $(Y,d_Y)$ two metric spaces (def. 1), then a function
is said to be continuous at a point $x \in X$ if for every $\epsilon \gt 0$ there exists $\delta\gt 0$ such that
or equivalently such that
where $B^\circ$ denotes the open ball (definition 2).
The function $f$ is called just continuous if it is continuous at every point $x \in X$.
This definition is equivalent to a more abstract one, which does not explicitly refer to points or radii anymore:
Let $(X,d)$ be a metric space (def. 1). Say that
A neighbourhood of a point $x \in X$ is a subset $x \in U \subset X$ which contains some open ball $B_x^\circ(\epsilon)$ around $x$ (def. 2).
An open subset of $X$ is a subset $U \subset X$ such that for every for $x \in U$ it also contains a neighbourhood of $x$.
The collection of open subsets in def. 4 constitutes a topology on the set $X$, making it a topological space. This is called the metric topology. Stated more concisely: the open balls in a metric space constitute the basis of a topology for the metric topology.
A function $f \colon X \to Y$ between metric spaces (def. 1) is continuous in the epsilontic sense of def. 3 precisely if it has the property that its pre-images of open subsets of $Y$ (in the sense of def. 4) are open subsets of $X$.
First assume that $f$ is continuous in the epsilontic sense. Then for $O_Y \subset Y$ any open subset and $x \in f^{-1(O_Y)}$ any point in the pre-image, we need to show that there exists a neighbourhood of $x$ in $U$. But by assumption there exists an open ball $B_x^\circ(\epsilon)$ with $f(B_X^\circ(\epsilon)) \subset O_Y$. Since this is true for all $x$, by definition this means that $f^{-1}(O_Y)$ is open in $X$.
Conversely, assume that $f^{-1}$ takes open subsets to open subsets. Then for every $x \in X$ and $B_{f(x)}^\circ(\epsilon)$ an open ball around its image, we need to produce an open ball $B_x^\circ(\delta)$ in its pre-image. But by assumption $f^{-1}(B_{f(x)}^\circ(\epsilon))$ contains a neighbourhood of $x$ which by definition means that it contains such an open ball around $x$.
A function $f \;\colon\; X\to Y$ between topological spaces is a continuous map (or is said to be continuous) if for every open subset $U \subset Y$, the preimage $f^{-1}(U)$ is an open subset $X$.
In nonstandard analysis, this is equivalent to
A function $f \;\colon\; X\to Y$ between topological spaces is a continuous map (or is said to be continuous) if for every standard point? $x_1$ and every hyperpoint? $x_2$, if $x_1$ and $x_2$ are adequal? (infinitely close, or in other words if $x_2$ is in the halo of $x_1$), then $f(x_1)$ and $\multiscripts{^*}f{}(x_2)$ are adequal (where $\multiscripts{^*}f{}$ is the nonstandard extension? of $f$). Equivalently, $f$ is continuous iff $\multiscripts{^*}f{}$ is microcontinuous?.
A function $f$ between convergence spaces is continuous if for any filter $F$ such that $F \to x$, it follows that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the filterbase $\{F(A) \;|\; A \in F\}$.
A continuous map between locales is simply a frame homomorphism in the opposite direction. Equivalently (via the adjoint functor theorem), it may be defined as a homomorphism of inflattices whose left adjoint preserves finitary meets (and hence is a frame homomorphism).
Since continuity is defined in terms of preservation of property (namely preserving “openness” under preimages), it is natural to ask what other properties they preserve.
Also, when a property is not always preserved it is useful to label those maps which do preserve it for closer study.
By definition, the preimage of an open set is open.
Similarly, the preimage of an closed set is closed.
The image of a connected subset is again connected.
The image of a compact subset is again compact (see at continuous images of compact subsets are compact?)
The preimage of a compact set need not be compact; a continuous map for which this is true is known as a proper map.
The image of an open set need not be open; a continuous map for which this is true is said to be an open map. (Technically, an open map is any function with just this property.)
The image of an closed set need not be closed; a continuous map for which this is true is said to be an closed map. (Technically, a closed map is any function with just this property.)
A continuous map of topological spaces which is invertible as a function of sets is a homeomorphism if the inverse function is a continuous map as well.
Although these don’t make sense for arbitrary topological spaces (convergence spaces, locales, etc), they are special kinds of continuous maps in contexts such as metric spaces:
Various notions of continuous function are used in constructive mathematics. A function $f$ (say real-valued and defined on a real interval) is:
In classical mathematics, these are all equivalent when the domain is itself a closed and bounded interval, and all of them except for uniform continuity are equivalent in general. The same equivalences hold in intuitionistic mathematics, thanks to the fan theorem. But no two of these are equivalent in Russian constructivism.
In fact, assuming that $\mathbb{R}$ is defined as the set of located Dedekind cuts, there is the following negative result by Frank Waaldijk (Waaldijk2003): Without the fan theorem, there is no notion of continuity for set-theoretic functions in constructive mathematics, spelled “kontinuity” in the following, such that all of the following desiderata are met:
The key problem is that a uniformly continuous, positive-valued function defined on $[0,1]$ might fail to be bounded below by a positive number, since the interval $[0,1]$ might fail to be compact, yet its reciprocal (if also uniformly continuous) must be bounded above.
Waaldijk’s negative result can be circumvented by dropping the insistence on points and instead working with maps between locales, toposes, or formal spaces as studied in formal topology.