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
The empty space is the topological space with no points. That is, it is the empty set equipped with its unique topology.
The empty space is the (strict) initial object in TopologicalSpaces.
It satisfies all separation, compactness, and countability conditions (separability, first countability, second-countability). It is also both discrete and indiscrete, a distinction it shares only with the point space.
Debate rages over whether the empty space is connected (and also path-connected). With the common naive definitions that “a space is connected if it cannot be partitioned into two disjoint nonempty open subsets” and “a space is path-connected if any two points in it can be joined by a path,” the empty space is trivially both connected and path-connected.
However, in some ways these definitions are too naive. The question of whether the empty set is (path-)connected is analogous in many ways to the question of whether $1$ is prime. The above definitions are then analogous to saying that “a natural number $p$ is prime if any factor of it is either equal to $1$ or to $p$,” according to which $1$ is prime—but there are better definitions that exclude $1$.
For instance, we may say that “$p$ is prime if it has exactly two factors, itself and $1$;” with this definition $1$ is not prime, since it has exactly one factor. Likewise, we may say that a space is (path-)connected if it has exactly one (path-)component?; with this definition the empty space is not connected, since it has exactly zero components. (Lest you question that last statement, note that the correct definition of a (path-)component of a space $X$ is as an equivalence class of points of $X$ under some equivalence relation. There is a unique equivalence relation on the empty set, and it has zero equivalence classes.)
Here are some other reasons why the empty space should not be considered (path-)connected:
If the empty space were (path-)connected, unique decomposition into (path-)connected components would fail: $X \sqcup Y = \emptyset \sqcup X \sqcup Y = \dots$. This is analogous to how if $1$ were a prime, then unique factorization into primes would fail: $6 = 2 \cdot 3 = 1 \cdot 2 \cdot 3 = 1 \cdot 1 \cdot 2 \cdot 3 = \dots$.
In homotopy theory, one defines a space $X$ to be $k$-connected if $\pi_i(X)$ is trivial (that is, has exactly one element) for $i \le k$. When $k =0$ this says that $\pi_0(X)$ should have exactly one component—that is, that $X$ should be path-connected. (Actually, this definition really only makes sense if we phrase it in terms of homotopy groupoids; homotopy groups are only defined once we choose a basepoint, which is clearly impossible for the empty space.)
Category-theoretically, one may say that a space $X$ is connected if the functor $hom(X,-)$ preserves coproducts. Since $\hom(\emptyset,-)$ is constant at the point, it certainly does not preserve coproducts.
The statement that a product $X\times Y$ is connected if and only if its components $X$ and $Y$ are connected fails if the empty set is regarded as connected.
A general result, e.g., in the theory of combinatorial species, is that the logarithm of exponential generating functions of some type of objects should be the exponential generating function for the connected objects of that type. Since this logarithm has no constant term, this suggests the empty object should not count as connected. This result is also known in the physics literature as the linked-cluster theorem (see this Prop. at geometry of physics – perturbative quantum field theory).
See too simple to be simple for general theory.
examples of universal constructions of topological spaces:
$\phantom{AAAA}$limits | $\phantom{AAAA}$colimits |
---|---|
$\,$ point space$\,$ | $\,$ empty space $\,$ |
$\,$ product topological space $\,$ | $\,$ disjoint union topological space $\,$ |
$\,$ topological subspace $\,$ | $\,$ quotient topological space $\,$ |
$\,$ fiber space $\,$ | $\,$ space attachment $\,$ |
$\,$ mapping cocylinder, mapping cocone $\,$ | $\,$ mapping cylinder, mapping cone, mapping telescope $\,$ |
$\,$ cell complex, CW-complex $\,$ |
Last revised on April 16, 2021 at 02:49:45. See the history of this page for a list of all contributions to it.