May
22
comment Is $\mathbb{Q}(\pi) \cong \mathbb{Q}[[x]]/ \langle \sin(x) \rangle$?
It may be worth noting that for any field $k$, $k[[x]]$ is a local ring with maximal ideal $(x)$. Thus, $\mathbb Q$ is the only field that can be expressed as a quotient of $\mathbb Q[[x]]$.
May
18
comment X a locally compact Haussdorf space, each singleton of which is the intersection of countably many open sets. Show that X is first countable.
possible duplicate of $G_\delta$ singletons in compact Hausdorff and first countability
May
16
answered Is it true that the intersection of a sequence $K_1 \supset K_2 \supset K_3 \dotsm$ of connected subsets of $\mathbb{R}^2$ is also connected?
May
15
revised Lee, Introduction to Smooth Manifolds Solutions
Removed link to pdf. Doesn't seem legally available and it's irrelevant in the context of the question.
May
14
comment Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then $A'$ is a closed set.
@Alyssa Exactly.
May
14
answered Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then $A'$ is a closed set.
May
12
comment Why the set $g^{-1}(\{0\}) $ is not a differentiable manifold?
This set is not even a topological manifold. Removing $(0, 0)$ leaves $4$ connected components.
May
10
comment Proposition 2.24 on Liu Qing‘s AG book
If $f$ is a closed immersion, then $f(Y)$ is closed by definition. It's unclear what you're asking.
May
10
comment Show that if $X$ and $Y$ are regular, then so is the product space $X\times Y$.
This answer assumes that all closed subsets of $X \times Y$ have the form $A \times B$. This isn't true in general.
May
9
answered $\int^b_a t^kf(t) dt\,=0$ for all $k \geq 1 \implies f=0$ a.e.
May
8
comment Help show the following isomorphism cannot exist.
The polynomial ring is $\mathbb Z_3[x]$ in the test you linked to.
May
4
comment Describe the closure in the Zariski topology
Do you know how to show that $Z(x^2 - y)$ as a variety is isomorphic to the affine line?
May
3
comment Describe the closure in the Zariski topology
That doesn't make sense. Note that $Z(x^2 - y)$ is homeomorphic to $\operatorname{Spec} k[x, y] / (x^2 - y) \cong \mathbb A_k^1$. As @GregoryGrant pointed out, since all proper closed subsets of $\mathbb A_k^1$ are finite, it follows that $Z(x^2 - y)$ is the smallest closed subset that contains $C$ (an infinite set).
May
3
comment Describe the closure in the Zariski topology
Re 2, $\mathbb C^2$ is irreducible. The given subset is open, so it must be dense.
May
3
comment Computing the homology groups of a quotient space of the sphere
$H_1(X)$ is not trivial. How did you do your computation?
May
2
comment when a presheaf is a sheaf
Do you mean the categorical definition using an equalizer diagram?
May
2
comment Approximating a CW-complex with a trivial fundamental group
@user2715119 I recommend looking it up in Hatcher's book. It's a useful tool for this kind of problems.
May
2
revised Approximating a CW-complex with a trivial fundamental group
More searchable title and better TeX.
May
2
answered Approximating a CW-complex with a trivial fundamental group
May
1
revised Science Bowl Question Regarding Connecting Segments
edited tags
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