topology (point-set topology, point-free topology)
see also differential topology, 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
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
A natural topology on mapping spaces of continuous functions, important because of its role in exhibiting locally compact topological spaces to be exponentiable, as demonstrated below, culminating in Corollary 1.
The compact-open topology on the set of continuous functions $X \to Y$ is generated by the subbasis of subsets $U^K \subset C(X,Y)$ that map a given compact subspace $K \subset X$ to a given open subset $U \subset Y$, whence the name.
When restricting to continuous functions between compactly generated topological spaces one usually modifies this definition to a subbase of open subsets $U^{\phi(K)}$, where now $\phi(K)$ is the image of a compact topological space under any continuous function $\phi \colon K \to X$. This definition gives a cartesian internal hom in the category of compactly generated topological spaces (see also at convenient category of topological spaces).
The two definitions agree when the domain $X$ is a compactly generated Hausdorff space, but not in general. Beware that texts on compactly generated spaces nevertheless commonly say “compact-open topology” for the second definition.
Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be topological spaces. We denote by $Y^{X}$ the set of continuous maps from $(X,\mathcal{O}_{X})$ to $(Y, \mathcal{O}_{Y})$.
Other common notations for $Y^{X}$ are $Map(X,Y)$ or $C(X,Y)$.
Let $(X, \mathcal{O}_{X})$ be a topological space. We denote by $\mathcal{O}^{c}_{X}$ the set of subsets of $X$ which are compact with respect to $\mathcal{O}_{X}$.
Let $(X, \mathcal{O}_{X})$ be a topological space. Let $A$ be a subset of $X$. We denote by $\overline{A}$ the topological closure of $A$ in $X$ with respect to $\mathcal{O}_{X}$.
A topological space $(X, \mathcal{O}_{X})$ is locally compact if the set of compact neighbourhoods of points of $X$ form a neighbourhood basis of $X$. That is to say: for every $x \in X$ and every neighbourhood $N$ of $x$, there is a $V \in \mathcal{O}^{c}_{X}$ and a $U \in \mathcal{O}_{X}$ such that $x \in U \subset V \subset N$.
There are many variations on this definition, which can be found at locally compact space. These are all equivalent if $(X, \mathcal{O}_{X})$ is Hausdorff. We do not however make the assumption that $(X, \mathcal{O}_{X})$ is Hausdorff.
Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be topological spaces. Given $A \in \mathcal{O}^{c}_{X}$ and $U \in \mathcal{O}_{Y}$, we denote by $M_{A,U}$ the set of continuous maps $f : X \rightarrow Y$ such that $f(A) \subset U$.
Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be topological spaces. The compact-open topology on $Y^{X}$ is that with sub-basis given by the set of sets $M_{A,U}$ such that $A \in \mathcal{O}^{c}_{X}$ and $U \in \mathcal{O}_{Y}$.
Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be topological spaces. We shall denote the compact-open topology (def. 2) on $Y^{X}$ by $\mathcal{O}_{Y^{X}}$.
Let $(X, \mathcal{O}_{X})$ be a locally compact topological space, def. 1, and let $(Y, \mathcal{O}_{Y})$ be a topological space. The map $ev : X \times Y^{X} \rightarrow Y$ given by $(x,f) \mapsto f(x)$ is continuous, where $X \times Y^{X}$ is equipped with the product topology $\mathcal{O}_{X \times Y^{X}}$ with respect to $\mathcal{O}_{X}$ and $\mathcal{O}_{Y^{X}}$ (notation 5).
By an elementary fact concerning continuous maps, it suffices to show that for any $(x,f) \in X \times Y^{X}$, and any $U \in \mathcal{O}_{Y}$ such that $f(x) \in U$, there is a $U' \in \mathcal{O}_{X \times Y^{X}}$ such that $(x,f) \in U'$, and such that $ev(U') \subset U$.
To demonstrate this, we make the following observations.
1) Since $f$ is continuous, we have that $f^{-1}(U) \in \mathcal{O}_{X}$.
2) Since $(X, \mathcal{O}_{X})$ is locally compact, we deduce from 1) that there is a $V \in \mathcal{O}^{c}_{X}$ and a $U'' \in \mathcal{O}_{X}$ such that $x \in U'' \subset V \subset f^{-1}(U)$.
3) By 2) and by definition of $\mathcal{O}_{X \times Y^{X}}$ and $\mathcal{O}_{Y^{X}}$, we have that $U'' \times M_{V, U} \in \mathcal{O}_{X \times Y^{X}}$, and that $(x,f) \in U'' \times M_{V, U}$.
4) Let $(x',f') \in U'' \times M_{V, U}$, Since $x' \in U''$, and since $f'(U'') \subset f'(V) \subset U$, the latter inclusion holding by definition of $M_{V, U}$, we have that $f'(x) \in U$. We deduce that $ev(V \times M_{ V, U}) \subset U$.
By 3) and 4), we see that we can take $U'$ as the beginning of the proof to be $U'' \times M_{V , U}$.
Let $(X, \mathcal{O}_{X})$, $(Y, \mathcal{O}_{Y})$, and $(Z, \mathcal{O}_{Z})$ be topological spaces. Let $f : X \times Y \rightarrow Z$ be a continuous map, where $X \times Y$ is equipped with the product topology $\mathcal{O}_{X \times Y}$ with respect to $\mathcal{O}_{X}$ and $\mathcal{O}_{Y}$. Then the map $\alpha_{f} : X \rightarrow Z^{Y}$ given by $x \mapsto f_{x}$ is continuous, where $f_{x} : Y \rightarrow Z$ is given by $y \mapsto f(x,y)$, and where $Z^{Y}$ is equipped with the compact-open topology $\mathcal{O}_{Z^{Y}}$.
By an elementary fact concerning continuous maps, it suffices to show that for any $x \in X$, and any $M_{A,U} \in \mathcal{O}_{Z^{Y}}$ such that $f_{x} \in M_{A,U}$, there is a $U' \in \mathcal{O}_{X}$ such that $x \in U'$, and such that $\alpha_{f}( U' ) \subset M_{A,U}$.
To demonstrate this, we make the following observations.
1) Since $f_{x} \in M_{A,U}$, we have, by definition of $f_{x}$ and by definition of $M_{A,U}$, that $f(x,a) \in U$ for all $a \in A$.
2) Since $f$ is continuous, we have that $f^{-1}(U) \in \mathcal{O}_{X \times Y}$.
3) By 1), we have that $\{x\} \times A \subset f^{-1}(U)$.
4) Let $\mathcal{O}_{X \times A}$ denote the subspace topology on $X \times A$ with respect to $\mathcal{O}_{X \times Y}$. By 2), we have that $f^{-1}(U) \cap (X \times A) \in \mathcal{O}_{X \times A}$.
5) By 3), we have that $\{x \} \times A \subset f^{-1}(U) \cap (X \times A)$.
6) Since $A \in \mathcal{O}^{c}_{Y}$, it follows from 4), 5), and the tube lemma that there is a $U' \in \mathcal{O}_{X}$ such that $x \in U'$ and such that $U' \times A \subset f^{-1}(U) \cap (X \times A) \subset f^{-1}(U)$.
7) We deduce from 6) that $f(U' \times A) \subset U$. This is the same as to say that $f_{x'}(a) \in U$ for all $x' \in U'$. Thus $f_{x'}$ belongs to $M_{A,U}$ for all $x' \in U'$, which is the same as to say that $\alpha_{f}(U') \subset M_{A,U}$.
We conclude that we can take the required $U'$ of the beginning of the proof to be the $U'$ of 6).
Let $(X, \mathcal{O}_{X})$, $(Y, \mathcal{O}_{Y})$, and $(Z, \mathcal{O}_{Z})$ be topological spaces. Suppose that $(X, \mathcal{O}_{X})$ is locally compact. Let $f : X \rightarrow Z^{Y}$ be a continuous map, where $Z^{Y}$ is equipped with the compact-open topology $\mathcal{O}_{Z^{Y}}$. Then the map $\alpha^{-1}_{f} : X \times Y \rightarrow Z$ given by $(x,y) \mapsto \big( f(x) \big)(y)$ is continuous, where $X \times Y$ is equipped with the product topology $\mathcal{O}_{X \times Y}$.
We make the following observations.
1) We have that $\alpha^{-1}_{f} = ev \circ \tau \circ (f \times id)$, where $\tau : Z^{Y} \times Y \rightarrow Y \times Z^{Y}$ is given by $(f,y) \mapsto (y,f)$.
2) By Proposition 2, we have that $ev$ is continuous.
3) It is an elementary fact that $\tau$ is continuous.
4) Since $f$ is continuous, it follows by an elementary fact that $f \times id$ is continuous.
5) We deduce from 2) - 4) that $ev \circ \tau \circ (f \times id)$ is continuous. By 1), we conclude that $\alpha^{-1}_{f}$ is continuous.
Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be topological spaces. Suppose that $(X, \mathcal{O}_{X})$ is locally compact. Then $(Y^{X}, \mathcal{O}_{Y^{X}})$ together with the corresponding map $ev$ defines an exponential object in the category Top of all topological spaces.
Follows immediately from Proposition 2, Proposition 3, and the fact that $Y^{X}$ and $ev$ are exhibited by the corresponding maps $\alpha_{-}$ and $\alpha^{-1}_{-}$ of Proposition 2 and Proposition 3 to define an exponential object in the category $\mathsf{Set}$ of sets.
A proof can also be found in Aguilar-Gitler-Prieto 02, prop. 1.3.1, or just about any half-decent textbook on point-set topology!
However, the result is almost universally stated with an assumption that $(X,\mathcal{O}_{X})$ is Hausdorff which, as the proof we have given illustrates, is not needed.
We moreover have a homeomorphism $Map(X,Map(Y,Z))\cong Map(X\times Y,Z)$ if in addition $X$ is Hausdorff. See also convenient category of topological spaces.
(mapping space construction out of the point space is the identity)
The point space $\ast$ is clearly a locally compact topological space. Hence for every topological space $(X,\tau)$ the mapping space $Maps(\ast, (X,\tau))$ exists. This is homeomorphic to the space $(x,\tau)$ itself:
If $Y$ is a metric space then the compact-open topology on $Map(X,Y)$ is the topology of uniform convergence on compact subsets in the sense that $f_n \to f$ in $Map(X,Y)$ with the compact-open topology iff for every compact subset $K\subset X$, $f_n \to f$ uniformly on $K$. If (in addition) the domain $X$ is compact then this is the topology of uniform convergence.
(loop space and path space)
Let $(X,\tau)$ be any topological space.
The standard circle $S^1$ is a compact Hausdorff space hence a locally compact topological space. Accordingly the mapping space
exists. This is called the free loop space of $(X,\tau)$.
If both $S^1$ and $X$ are equipped with a choice of point (“basepoint”) $s_0 \in S^1$, $x_0 \in X$, then the topological subspace
on those functions which take the basepoint of $S^1$ to that of $X$, is called the loop space of $X$, or sometimes based loop space, for emphasis.
Similarly the closed interval is a compact Hausdorff space hence a locally compact topological space (def. \ref{LocallyCompactSpace}). Accordingly the mapping space
exists. Again if $X$ is equipped with a choice of basepoint $x_0 \in X$, then the topological subspace of those functions that take $0 \in [0,1]$ to that chosen basepoint is called the path space of $(X\tau)$:
Notice that we may encode these subspaces more abstractly in terms of universal properties:
The path space and the loop space are characterized, up to homeomorphisms, as being the limiting cones in the following pullback diagrams of topological spaces:
Here on the right we are using that the mapping space construction is a functor and we are using example 1 in the identification on the bottom right mapping space out of the point space.
Ralph H. Fox, On Topologies for Function Spaces , Bull. AMS 51 (1945) pp.429-432. (pdf)
Eva Lowen-Colebunders, Günther Richter, An Elementary Approach to Exponential Spaces, Applied Categorical Structures May 2001, Volume 9, Issue 3, pp 303-310 (publisher)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, sections 1.2, 1.3 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)