That paracompact Hausdorff implies normal is standard and there are examples on StackExchange of perfectly normal Hausdorff spaces that are not paracompact, but I'm not sure of the answer, especially since paracompact spaces are collection-wise normal and the latter is not related to perfectly normal. (The standard example of a collection-wise normal space that is not perfectly normal is $\omega_1$, which is not paracompact.)
2026-02-23 01:02:48.1771808568
Does paracompact Hausdorff imply perfectly normal?
391 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in GENERAL-TOPOLOGY
- Is every non-locally compact metric space totally disconnected?
- Let X be a topological space and let A be a subset of X
- Continuity, preimage of an open set of $\mathbb R^2$
- Question on minimizing the infimum distance of a point from a non compact set
- Is hedgehog of countable spininess separable space?
- Nonclosed set in $ \mathbb{R}^2 $
- I cannot understand that $\mathfrak{O} := \{\{\}, \{1\}, \{1, 2\}, \{3\}, \{1, 3\}, \{1, 2, 3\}\}$ is a topology on the set $\{1, 2, 3\}$.
- If for every continuous function $\phi$, the function $\phi \circ f$ is continuous, then $f$ is continuous.
- Defining a homotopy on an annulus
- Triangle inequality for metric space where the metric is angles between vectors
Related Questions in EXAMPLES-COUNTEREXAMPLES
- A congruence with the Euler's totient function and sum of divisors function
- Seeking an example of Schwartz function $f$ such that $ \int_{\bf R}\left|\frac{f(x-y)}{y}\right|\ dy=\infty$
- Inner Product Uniqueness
- Metric on a linear space is induced by norm if and only if the metric is homogeneous and translation invariant
- Why do I need boundedness for a a closed subset of $\mathbb{R}$ to have a maximum?
- A congruence with the Euler's totient function and number of divisors function
- Analysis Counterexamples
- A congruence involving Mersenne numbers
- If $\|\ f \|\ = \max_{|x|=1} |f(x)|$ then is $\|\ f \|\ \|\ f^{-1}\|\ = 1$ for all $f\in \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$?
- Unbounded Feasible Region
Related Questions in SEPARATION-AXIOMS
- A finite topological space is T1 if and only the topology is discrete
- normal doesn't imply paracompact
- Find the intersection of all $T_2$ topologies on an infinite set $X$
- Why isn't $T_1$ space also Hausforff?
- Quotient space and quotient set for $\mathrm{End}(\mathbb{R}^2)$
- $\left\{e\right\}$ Closed Implies $T_0$ and Regular Topological Group
- Question about the proof of $T_3+$ countable basis $= T_4$
- Show that a non-trivial connected Tychonoff space is uncountable
- A homeomorphism on a dense set in Hausdorff space
- Unclear definition of open sets in proof of Regular Lindelof space is normal
Related Questions in PARACOMPACTNESS
- normal doesn't imply paracompact
- Space of Sequences with Finitely Many Nonzero Terms is Paracompact
- Theorem 41.7 in Munkres Topology
- Tietze extension theorem for vector bundles on paracompact spaces
- Proof a theorem about Metrizable manifold
- Every $F_\sigma$-set in a paracompact space is paracompact.
- Uncountable product of many copies of $\mathbb{Z}$ is not paracompact
- Construct Compact Exhaustion using Paracompactness
- Locally compact topological group is paracompact
- A cover of Locally connected space with certain compactness property
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
No. Consider the closed ordinal space $\omega_1+1 = [0,\omega_1]$. Since it is compact, it is clearly paracompact, and is also Hausdorff. But it is not perfectly normal: the (closed) singleton $\{ \omega_1 \}$ is not a Gδ subset.