Is standard topology normal?
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A normal space is a space (typically a topological space) which satisfies one of the stronger separation axioms. Show DefinitionsA topological space XX is normal if it satisfies: By Urysohn's lemma this is equivalent to the condition Often one adds the requirement
(Unlike with regular spaces, T 0T_0 is not sufficient here.) One may also see terminology where a normal space is any space that satisfies T 4T_4, while a T 4T_4-space must satisfy both T 4T_4 and T 1T_1. This has the benefit that a T 4T_4-space is always also a T 3T_3-space while still having a term available for the weaker notion. On the other hand, the reverse might make more sense, since you would expect any space that satisfies T 4T_4 to be a T 4T_4-space; this convention is also seen. If instead of T 1T_1, one requires
then the result may be called an R 3R_3-space. Any space that satisfies both T 4T_4 and T 1T_1 must be Hausdorff, and every Hausdorff space satisfies T 1T_1, so one may call such a space a normal Hausdorff space; this terminology should be clear to any reader. Any space that satisfies both T 4T_4 and R 0R_0 must be regular (in the weaker sense of that term), and every regular space satisfies R 0R_0, so one may call such a space a normal regular space; however, those who interpret ‘normal’ to include T 1T_1 usually also interpret ‘regular’ to include T 1T_1, so this term can be ambiguous. Every normal Hausdorff space is an Urysohn space, a fortiori regular and a fortiori Hausdorff. It can be useful to rephrase T 4T_4 in terms of only open sets instead of also closed ones:
This definition is suitable for generalisation to locales and also for use in constructive mathematics (where it is not equivalent to the usual one). To spell out the localic case, a normal locale is a frame LL such that
Examples
We need to show that given two disjoint closed subsets C 1,C 2⊂XC_1, C_2 \subset X, there exist disjoint open neighbourhoods U C 1⊃C 1U_{C_1} \supset C_1 and U C 2⊃C 2U_{C_2} \supset C_2. Consider the function d(S,−):X→ℝ d(S,-) \colon X \to \mathbb{R} which computes distances from a subset S⊂XS \subset X, by forming the infimum of the distances to all its points: d(S,x)≔inf{d(s,x)|s∈S}. d(S,x) \coloneqq inf\left\{ d(s,x) \vert s \in S \right\} \,. If SS is closed and x∉Sx \notin S, then d(S,x)>0d(S, x) \gt 0. Then the unions of open balls U C 1≔⋃x 1∈C 1B x 1 ∘(12d(C 2,x 1)) U_{C_1} \coloneqq \underset{x_1 \in C_1}{\bigcup} B^\circ_{x_1}( \frac1{2}d(C_2,x_1) ) and U C 2≔⋃x 2∈C 2B x 2 ∘(12d(C 1,x 2)). U_{C_2} \coloneqq \underset{x_2 \in C_2}{\bigcup} B^\circ_{x_2}( \frac1{2}d(C_1,x_2) ) \,. have the required properties. For if there exist x 1∈C 1,x 2∈C 2x_1 \in C_1, x_2 \in C_2 and y∈B x 1 ∘(12d(C 2,x 1))∩B x 2 ∘(12d(C 1,x 2))y \in B^\circ_{x_1}( \frac1{2}d(C_2,x_1) ) \cap B^\circ_{x_2}( \frac1{2}d(C_1,x_2) ), then d(x 1,x 2)≤d(x 1,y)+d(y,x 2)<12(d(C 2,x 1)+d(C 1,x 2))≤max{d(C 2,x 1),d(C 1,x 2)}d(x_1, x_2) \leq d(x_1, y) + d(y, x_2) \lt \frac1{2} (d(C_2, x_1) + d(C_1, x_2)) \leq \max\{d(C_2, x_1), d(C_1, x_2)\} and if d(C 1,x 2)≤d(C 2,x 1)d(C_1, x_2) \leq d(C_2, x_1) say, then d(x 2,x 1)=d(x 1,x 2)
Every regular second countable space is normal. See Urysohn metrization theorem for details. See paracompact Hausdorff spaces are normal for details.
By construction the K-topology is finer than the usual euclidean metric topology. Since the latter is Hausdorff, so is ℝ K\mathbb{R}_K. It remains to see that ℝ K\mathbb{R}_K contains a point and a disjoint closed subset such that they do not have disjoint open neighbourhoods. But this is the case essentially by construction: Observe that ℝ\K=(−∞,−1/2)∪((−1,1)\K)∪(1/2,∞) \mathbb{R} \backslash K \;=\; (-\infty,-1/2) \cup \left( (-1,1) \backslash K \right) \cup (1/2, \infty) is an open subset in ℝ K\mathbb{R}_K, whence K=ℝ\(ℝ\K) K = \mathbb{R} \backslash ( \mathbb{R} \backslash K ) is a closed subset of ℝ K\mathbb{R}_K. But every open neighbourhood of {0}\{0\} contains at least (−ϵ,ϵ)\K(-\epsilon, \epsilon) \backslash K for some positive real number ϵ\epsilon. There exists then n∈ℕ ≥0n \in \mathbb{N}_{\geq 0} with 1/n<ϵ1/n \lt \epsilon and 1/n∈K1/n \in K. An open neighbourhood of KK needs to contain an open interval around 1/n1/n, and hence will have non-trivial intersection with (−ϵ,ϵ)(-\epsilon, \epsilon). Therefore {0}\{0\} and KK may not be separated by disjoint open neighbourhoods, and so ℝ K\mathbb{R}_K is not normal.
If ω 1\omega_1 is the first un-countable ordinal with the order topology, and ω 1¯\widebar{\omega_1} its one-point compactification, then X=ω 1×ω 1¯X = \omega_1 \times \widebar{\omega_1} with the product topology is not normal. Indeed, let ∞∈ω 1¯\infty \in \widebar{\omega_1} be the unique point in the complement of ω 1↪ω 1¯\omega_1 \hookrightarrow \widebar{\omega_1}; then it may be shown that every open set UU in XX that includes the closed set A={(x,x):x≠∞}A = \{(x, x): x \neq \infty\} in XX must somewhere intersect the closed set ω 1×{∞}\omega_1 \times \{\infty\} which is disjoint from AA. For if that were false, then we could define an increasing sequence x n∈ω 1x_n \in \omega_1 by recursion, letting x 0=0x_0 = 0 and letting x n+1∈ω 1x_{n+1} \in \omega_1 be the least element that is greater than x nx_n and such that (x n,x n+1)∉U(x_n, x_{n+1}) \notin U. Then, letting b∈ω 1b \in \omega_1 be the supremum of this increasing sequence, the sequence (x n,x n+1)(x_n, x_{n+1}) converges to (b,b)(b, b), and yet the neighborhood UU of (b,b)(b, b) contains none of the points of this sequence, which is a contradiction. This example also shows that general subspaces of normal spaces need not be normal, since ω 1×ω 1¯\omega_1 \times \widebar{\omega_1} is an open subspace of the compact Hausdorff space ω 1¯×ω 1¯\widebar{\omega_1} \times \widebar{\omega_1}, which is itself normal.
An uncountable product of infinite discrete spaces XX is not normal. More generally, a product of T 1T_1 spaces X iX_i uncountably many of which are not limit point compact is not normal. Indeed, by a simple application of Remark below and the fact that closed subspaces of normal Hausdorff spaces are normal Hausdorff, it suffices to see that the archetypal example ℕ ω 1\mathbb{N}^{\omega_1} is not normal. For a readable and not overly long account of this result, see Dan Ma’s blog. PropertiesBasic properties
(normality in terms of topological closures) A topological space (X,τ)(X,\tau) is normal Hausdorff, precisely if all points are closed and for all closed subsets C⊂XC \subset X with open neighbourhood U⊃CU \supset C there exists a smaller open neighbourhood V⊃CV \supset C whose topological closure Cl(V)Cl(V) is still contained in UU: C⊂V⊂Cl(V)⊂U. C \subset V \subset Cl(V) \subset U \,.
In one direction, assume that (X,τ)(X,\tau) is normal, and consider C⊂U. C \subset U \,. It follows that the complement of the open subset UU is closed and disjoint from CC: C∩X∖U=∅. C \cap X \setminus U = \emptyset \,. Therefore by assumption of normality of (X,τ)(X,\tau), there exist open neighbourhoods with V⊃C,AAW⊃X∖UAAwithAAV∩W=∅. V \supset C \,, \phantom{AA} W \supset X \setminus U \phantom{AA} \text{with} \phantom{AA} V \cap W = \emptyset \,. But this means that V⊂X∖W V \subset X \setminus W and since the complement X∖WX \setminus W of the open set WW is closed, it still contains the closure of VV, so that we have C⊂V⊂Cl(V)⊂X∖W⊂U C \subset V \subset Cl(V) \subset X \setminus W \subset U as required. In the other direction, assume that for every open neighbourhood U⊃CU \supset C of a closed subset CC there exists a smaller open neighbourhood VV with C⊂V⊂Cl(V)⊂U. C \subset V \subset Cl(V) \subset U \,. Consider disjoint closed subsets C 1,C 2⊂X,AAAC 1∩C 2=∅. C_1, C_2 \subset X \,, \phantom{AAA} C_1 \cap C_2 = \emptyset \,. We need to produce disjoint open neighbourhoods for them. From their disjointness it follows that X∖C 2⊃C 1 X \setminus C_2 \supset C_1 is an open neighbourhood. Hence by assumption there is an open neighbourhood VV with C 1⊂V⊂Cl(V)⊂X∖C 2. C_1 \subset V \subset Cl(V) \subset X \setminus C_2 \,. Thus V⊃C 1,AAAAX∖Cl(V)⊃C 2 V \supset C_1 \,, \phantom{AAAA} X \setminus Cl(V) \supset C_2 are two disjoint open neighbourhoods, as required.
The separation conditions T 0T_0 to T 4T_4 may equivalently be understood as lifting properties against certain maps of finite topological spaces, among others. This is discussed at separation axioms in terms of lifting properties, to which we refer for further details. Here we just briefly indicate the corresponding lifting diagrams. In the following diagrams, the relevant finite topological spaces are indicated explicitly by illustration of their underlying point set and their open subsets:
In the lifting diagrams for T 2−T 4T_2-T_4 below, an arrow out of the given topological space XX is a map that determines (classifies) a decomposition of XX into a union of subsets with properties indicated by the picture of the finite space. Notice that the diagrams for T 2T_2-T 4T_4 below do not in themselves imply T 1 T_1 . Tietze extension and lifting propertyThe Tietze extension theorem applies to normal spaces. In fact the Tietze extension theorem can serve as a basis of a category theoretic characterization of normal spaces: a (Hausdorff) space XX is normal if and only if every function f:A→ℝf \colon A \to \mathbb{R} from a closed subspace A⊂XA \subset X admits an extension f˜:X→ℝ\tilde{f}: X \to \mathbb{R}, or what is the same, every regular monomorphism into XX in HausHaus has the left lifting property with respect to the map ℝ→1\mathbb{R} \to 1. (See separation axioms in terms of lifting properties (Gavrilovich 14) for further categorical characterizations of various topological properties in terms of lifting problems.) The category of normal spacesAlthough normal (Hausdorff) spaces are “nice topological spaces” (being for example Tychonoff spaces, by Urysohn's lemma), the category of normal topological spaces with continuous maps between them seems not to be very well-behaved. (Cf. the rule of thumb expressed in dichotomy between nice objects and nice categories.) It admits equalizers of pairs of maps f,g:X⇉Yf, g: X \rightrightarrows Y (computed as in TopTop or HausHaus; one uses the easy fact that closed subspaces of normal spaces are normal). However it curiously does not have products – or at least it is not closed under products in TopTop, as shown by Counter-Example . It follows that this category is not a reflective subcategory of TopTop, as HausHaus is. More at the page colimits of normal spaces. ReferencesThe class of normal spaces was introduced by Tietze (1923) and Aleksandrov–Uryson (1924).
Discussion of separation axioms in terms of lifting properties is due to: |
