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Chapter 1: Problem 10

(a) If \(Y\) is any subset of a topological space \(X,\) then \(\operatorname{dim} Y \leqslant \operatorname{dim} X\) (b) If \(X\) is a topological space which is covered by a family of open subsets \(\left\\{U_{i}\right\\}\) then \(\operatorname{dim} X=\sup \operatorname{dim} U_{i}\) (c) Give an example of a topological space \(X\) and a dense open subset \(U\) with \(\operatorname{dim} U<\operatorname{dim} X\) (d) If \(Y\) is a closed subset of an irreducible finite-dimensional topological space \(X\) and if \(\operatorname{dim} Y=\operatorname{dim} X,\) then \(Y=X\) (e) Give an example of a noetherian topological space of infinite dimension.

Short Answer

Expert verified
For part (a), the dimension of a subset is always less than or equal to the dimension of the full set. For part (b), the dimension of a topological space equals the supremum of the dimensions of its open subsets. Illustratively, a disk in a plane for part (c) which is dense in the plane but has lower dimension than the plane. If $Y$ is a subset of finite dimensional irreducible space $X$ and dim($Y$)=dim($X$) then $Y=X$ in part (d). Lastly for (e), the space of all algebraic numbers serves as an example of a noetherian topological space of infinite dimension.

Step by step solution

01

Solution for part (a)

Assume $Y$ is a subset of topological space $X$. The dimension of a topological space is the supremum of the dimensions of its closed subsets. Since each closed subset of $Y$ is also a closed subset of $X$, it must hold that $\operatorname{dim} Y \leqslant \operatorname{dim} X$.
02

Solution for part (b)

Let $\left\{U_{i}\right\}$ be a family of open subsets of topological space $X$. $X$ covered by these subsets means that for each point $x \in X$, there exists an $U_{i}$ such that $x$ belongs to $U_{i}$. The dimension of a topological space is equal to the supremum of the dimensions of its open subsets, thus $\operatorname{dim} X = \sup \operatorname{dim} U_{i}$.
03

Solution for part (c)

Consider the Euclidean space $\mathbb{R}^{2}$, and let $U$ be any dense open subset such as an open disk. The dimension of $U$ is 1, which is less than the dimension of $\mathbb{R}^{2}$, which is 2.
04

Solution for part (d)

Given that $Y$ is a closed subset of an irreducible finite-dimensional topological space $X$ and $\operatorname{dim} Y=\operatorname{dim} X$. Because $X$ is irreducible, then given any set $Z$ not equal to $X$, we have $\operatorname{dim} Z<\operatorname{dim} X$. Hence $Y=X$ for $\operatorname{dim} Y = \operatorname{dim} X$.
05

Solution for part (e)

One example of a noetherian topological space of infinite dimension is the space of all algebraic numbers. Each number is closed in its own right, and the infinite ascending chain of subsets defines an infinite dimension.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Topological Space
A topological space is a fundamental concept in topology, a branch of mathematics concerned with the notion of space and continuous transformations. A topological space consists of a set of points, along with a collection of open sets that satisfy certain properties. These properties are:
  • The empty set and the entire set must be open.
  • The union of any collection of open sets must also be open.
  • The intersection of any finite collection of open sets must also be open.
These rules allow us to understand how subsets of a space relate to each other and which transformations preserve these relationships. Topological spaces allow mathematicians to explore concepts of continuity, convergence, and compactness in a generalized form away from the confines of Euclidean spaces.
Dense Open Subset
A dense open subset of a topological space is an open set whose closure is the entire space. In simpler terms, this means that a dense open subset is close to being the entire space without actually including every point. For example, in the Euclidean space \(\mathbb{R}^2\) considered in the solution, any open set, such as an open disk, can be considered dense if its closure (the disk plus its boundary points) covers the entire space.

Dense open subsets are important because they help define the topological properties of the entire space. Though they might be missing a few points here and there, their reach throughout the space allows them to share many topological features with the entire set. This can be useful in various mathematical proofs and theories, particularly in understanding the behavior of functions and continuity.
Noetherian Space
A Noetherian space refers to a topological space satisfying the descending chain condition: every descending chain of open sets eventually becomes constant. This means you can't keep finding smaller and smaller open sets infinitely.

Noetherian spaces are often used in algebraic geometry and are named after the mathematician Emmy Noether. One interesting aspect of Noetherian spaces is that they are always quasi-compact, meaning every open cover has a finite subcover. An example of a Noetherian space from the solutions is the set of all algebraic numbers, which has an infinite topological dimension due to its inherent structure, yet satisfies this unique Noetherian condition. Noetherian spaces often exhibit properties that make them easier to manage within proofs of algebraic continuity and compactness.
Irreducible Space
Irreducible spaces are topological spaces that cannot be represented as the union of two proper closed subsets. In other words, every nonempty open subset of an irreducible space is dense, meaning it touches every corner of the space.

One of the key insights into irreducible spaces is that they are connected in a very strong way. They do not allow for disjoint closed portions within the space, as seen in the exercise solution where each set not equal to the space itself has a dimension less than the whole. This property makes irreducible spaces a central concept in the study of algebraic geometry and various mathematical analysis sectors.
  • It helps in understanding the structure of algebraic varieties.
  • It is instrumental in making certain proofs within topology simpler and more elegant.
Recognizing an irreducible space allows mathematicians to understand its deep, underlying connectedness, which is pivotal in various topological and geometrical contexts.

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Most popular questions from this chapter

Families of Plane Curves. A homogeneous polynomial \(f\) of degree \(d\) in three variables \(x, y, z\) has \(\left(\begin{array}{c}d+2 \\ 2\end{array}\right)\) coefficients. Let these coefficients represent a point in \(\mathbf{P}^{N},\) where \(N=\left(\begin{array}{c}d+2 \\ 2\end{array}\right)-1=\frac{1}{2} d(d+3).\) (a) Show that this gives a correspondence between points of \(\mathbf{P}^{N}\) and algebraic sets in \(\mathbf{P}^{2}\) which can be defined by an equation of degree \(d\). The correspondence is \(1-1\) except in some cases where \(f\) has a multiple factor. (b) Show under this correspondence that the (irreducible) nonsingular curves of degree \(d\) correspond \(1-1\) to the points of a nonempty Zariski-open subset of \(\mathbf{P}^{N} .[\text { Hints: }(1) \text { Use elimination theory }(5.7 \mathrm{A})\) applied to the homogeneous polynomials \(\partial f / \partial x_{0}, \ldots, \partial f / \partial x_{n} ;(2)\) use the previous (Ex. 5.5, 5.8, 5.9) above.

The Elliptic Quartic Curve in \(\mathbf{P}^{3}\). Let \(Y\) be the algebraic set in \(\mathbf{P}^{3}\) defined by the equations \(x^{2}-x z-y w=0\) and \(y z-x w-z w=0 .\) Let \(P\) be the point \((x, y, z, w)=(0,0,0,1),\) and let \(\varphi\) denote the projection from \(P\) to the plane \(w=0\). Show that \(\varphi\) induces an isomorphism of \(Y-P\) with the plane cubic curve \(y^{2} z-x^{3}+x z^{2}=0\) minus the point \((1,0,-1) .\) Then show that \(Y\) is an irreducible nonsingular curve. It is called the elliptic quartic curve in \(\mathbf{P}^{3}\). since it is defined by two equations it is another example of a complete intersection (Ex. 2.17 ).

Linear Varieties in \(\mathbf{P}^{n}\). A hypersurface defined by a linear polynomial is called a hyperplane (a) Show that the following two conditions are equivalent for a variety \(Y\) in \(\mathbf{P}^{\prime \prime}:\) (i) \(I(Y)\) can be generated by linear polynomials. (ii) \(Y\) can be written as an intersection of hyperplanes. In this case we say that \(Y\) is a linear rariety in \(\mathbf{P}^{\prime \prime}\) (b) If \(Y\) is a linear variety of dimension \(r\) in \(\mathbf{P}^{\prime \prime}\), show that \(I\) ( \(Y\) ) is minimally generated by \(n-r\) linear polynomials. (c) Let \(Y, Z\) be linear varieties in \(\mathbf{P}^{n},\) with \(\operatorname{dim} Y=r, \operatorname{dim} Z=s\). If \(r+s-n \geqslant 0\) then \(Y \cap Z \neq \varnothing\). Furthermore, if \(Y \cap Z \neq \varnothing\), then \(Y \cap Z\) is a linear variety of dimension \(\geqslant r+s-n .\) (Think of \(\mathbf{A}^{n+1}\) as a vector spacc over \(k\) and work with its subspaces.

Projective Closure of an Affine Varicty. If \(Y \subseteq \mathbf{A}^{n}\) is an affine variety, we identify \(\mathbf{A}^{n}\) with an open set \(U_{0} \subseteq \mathbf{P}^{n}\) by the homeomorphism \(\varphi_{0} .\) Then we can speak of \(\bar{Y},\) the closure of \(Y\) in \(\mathbf{P}^{n}\), which is called the projectice closure of \(Y\) (a) Show that \(I(\bar{Y})\) is the ideal generated by \(\beta(I(Y)),\) using the notation of the proof of (2.2) (b) Let \(Y \subseteq \mathbf{A}^{3}\) be the twisted cubic of (Ex. 1.2 ). Its projective closure \(\bar{Y} \subseteq \mathbf{P}^{3}\) is called the twisted cubic curce in \(\mathbf{P}^{3}\). Find generators for \(I(Y)\) and \(I(\bar{Y})\), and use this example to show that if \(f_{1}, \ldots, f_{r}\) generate \(I(Y),\) then \(\beta\left(f_{1}\right), \ldots, \beta\left(f_{r}\right)\) do not necessarily generate \(I(\bar{Y})\)

Let \(Y\) be the algebraic set in \(\mathbf{A}^{3}\) defined by the two polynomials \(x^{2}-y z\) and \(x z-x .\) Show that \(Y\) is a union of three irreducible components. Describe them and find their prime ideals.
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