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Questions tagged [general-topology]

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

57719 questions
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Do you have some hints as to how to solve the following problem?

Let $A_1,A_2,A_3,\ldots,A_n,\ldots$ be a collection of subsets in $[0,1]^{\Bbb N}$ and let $$A=A_1\times A_2\times A_3\times\ldots\subseteq[0,1]^{\Bbb N}\;.$$ Show…
jose maria
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Sets unchanged by a certain function form a topology?

Reviewing a bit of topology, and I'm trying to recall all the different ways to define a topology. Suppose I have some function $f$ on $\mathcal{P}(X)$ for some set $X$, that has the following properties. For any $S,T\subseteq…
DiD
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Is the subset $[0,1)$ of $\mathbb{R}$ compact in the lower limit topology?

What I have done so far is give a contradiction, namely the cover: $\mathcal{U}=\{{[0,1-\frac{1}{n}):n\in\mathbb{N}}\}$ Because $\cup_{n\in\mathbb{N}}[0,1-\frac{1}{n})=[0,1)$, it means that there is no finite subcover that covers $[0,1)$. Is this…
Gehaktmolen
  • 491
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Show that the cone of the open interval (0, 1) can not be embedded in any Euclidean space

I've been trying to tackle this problem for some while now, but don't know how to start correctly. I know that the cone on $(0,1)$ is given by $$\text{Cone}((0,1)) = (0,1) \times [0,1]/((0,1)\times\{1\}).$$ But how do I show that it can not be…
user112167
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determine basis for topology on $\mathbb{R}^2$

Determine whether the collection of subsets below form a basis for a topology on $\mathbb{R}^2$. All subsets of the form $T_{\epsilon}(x)=\lbrace (y_1,y_2) : |x_1+x_2-y_1-y_2| <\epsilon \rbrace$ for all $x \in \mathbb{R}^2$ and all $\epsilon>0$ By…
Idonknow
  • 15,643
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Show that for any open set $U$ in $\mathbb{R}^2$, $f(U)$ is an open set in $\mathbb{R}$?

Suppose $\mathbb{R}^2$ and $\mathbb{R}$ are topological spaces with standard topology. Let $f(x,y) = x + y^2$ . How do I show that for any open set $U$ in $\mathbb{R}^2$, $f(U)$ is an open set in $\mathbb{R}$? This looks similar to this…
sarah
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Help understand compact open topology

May I refer you to: Lemma 4.1.6 in: http://www.mathematik.tu-darmstadt.de/Math-Net/Lehrveranstaltungen/Lehrmaterial/SS2005/Liegruppen/skript/ch5vorl.pdf Page $4$ OK so I want to understand why that family forms a subbasis for the compact open…
user10
  • 5,688
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Elementary Borsuk-Ulam Proof Help

I'm working through "Elementary proof of Borsuk-Ulam Theorem" found here. Lemma 2: If there exists a continuous mapping of $f: \mathbb S^n \to \mathbb R^n$, which does not identify any pair of antipodes, then there exists an odd continuous…
Katrina Morrison
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Prove that the interior of the set of all orthogonal vectors to "a" is empty

I made a picture of the problem here: If the link does not work, read this: Let $a$ be a non-zero vector in $\mathbb{R}^n$. Let S be the set of all orthogonal vectors to $a$ in $\mathbb{R}^n$. I.e., for all $x \in \mathbb{R}^n$, $a\cdot x =…
user2193
  • 53
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Each closed set is $f^{-1}(0)$

Let $X$ be some compact Hausdorff space (or even normal space). Is it true that each closed subset $X'$ is $f^{-1}(0)$ for some $f\in C(X,\mathbb{R})$? I know that there is Urysohn's lemma which gives us an opportunity to continue each function…
user74574
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Given a countable partition of the reals, must the condensation points of some part have nonempty interior?

To say that $p$ is a condensation point of a subset $S$ of a topological space $X$ is to say that any open neighbourhood of $p$ contains uncountably many points of $S$. Let us suppose we have written $\mathbb{R} = \bigcup_{i=1}^\infty A_i$. I would…
Mike F
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Question asks for what I can say about the topologies of $T$ and $T'$ where $T'$ is strictly finer than $T$

Question: If $T$ and $T'$ are topologies on $X$ and $T'$ is strictly finer than $T$, what can you say about the corresponding subspace topologies on the subset $Y$ of $X$? I can never really know what more is required for questions like this. I…
Siyanda
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$X,Y$ topological spaces, $f:A\rightarrow Y$ continuous, $Y\rightarrow X\cup Y\rightarrow X\cup_f Y$ embedding

Let $X,Y$ both be topological spaces, $A\subseteq X$ and $f:A\rightarrow Y$ a continuous function. Take the disjoint union of $X$ and $Y$, $X\cup Y$, and then identify each $a \in A \subseteq X$ with $f(a) \in Y$ and take the quotient topology on…
user43138
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Prove that for every $n \geq 2$, $\Bbb R$ is not homeomorphic to $\Bbb R^n$

So for them to not be homeomorphic the function or inverse of the function must not be continuous. Correct? Should I assume homeomorphism first, and create open balls?
user105225
  • 133
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Prove that $\Bbb R^2 - \{0\}$ is homeomorphic to $S^1 \times \Bbb R$.

No idea where to even begin. There is a hint: this requires construction of an explicit function.
user105225
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