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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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The significance of Topologically Equivalence (To A Donut)?

A friend of mine explained how a coffee mug is topologically equivalent to a donut to me tonight. I have to say the idea is very interesting! However, I don't know topology a lot but I am wondering about the significance of such equivalence. Not to…
user48601
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prove that the closure of the intersection of A and B is the subset of the intersection of the closure of A and the closure of B.

prove that the closure of the intersection of A and B is the subset of the intersection of the closure of A and the closure of B. my proof: let x be in cl(A intersects B), then x is in the intersection of A and B and x is in the set of the limit of…
user110969
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Why second countable for definition of manifold?

What is the motivation to define a manifold to be second countable? What kind of pedagogical issues does this avoid?
LASV
  • 3,108
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Subnet vs. Subsequence

I'm looking for an example of a topological space $X$, a sequence $(x_n)_{n \in \mathbb{N}}$ in $X$ and a converging subnet $(x_i)_{i\in I}$ of $(x_n)$, but with the property that $x_n$ does not have any converging subsequence. I have an examples of…
user110894
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Open subsets of the closure

I want to prove that every open subset of a topological subpace is an open subset of its closure. Let $Y$ be a topological space and $X$ a subspace of $Y$. If $U$ is an open subset of $X$, we have $U=U\cap \overline X$, thus U is an open subset of…
user42912
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Is the intersection of a regular open with a dense subspace regular open in the subspace?

Let $X$ be a topological space (no assumptions about separation), $U$ be a regular open subset of $X$ (i.e. $\mbox{int}\,\mbox{cl}\, U = U$) and $D$ a dense subset of $X$. Is $D \cap U$ a regular open of $D$, i.e. is $\mbox{int}_D \mbox{cl}_D (D…
ihaphleas
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A quotient map from $[0,1]$ to $S^1$

I would like to show that the function $f(x) = (\textrm{cos}2 \pi x, \textrm{sin}2 \pi x)$ is a quotient map; I have already shown that it is surjective and continuous (the latter by invoking the universal property for functions into a product…
Bachmaninoff
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Infinite disjoint class of open subsets in an infinite Hausdorff space.

Let X be an infinite Hausdorff space. Prove that there exist an infinite disjoint class of open subsets of X. Ok the first time I tried to prove this I started by taking pairwise disjoint sets given by the Hausdorff property and then taking the…
pjox
  • 515
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What is a 0-ball?

I'm reading a paper that says $\bigcap V_{T,X}$ is either empty or a closed $l$-ball where $T \subset S$ is a subset of points $S$ and $\operatorname{card}{T} = m + 1 - l$ where $m$ is the dimension of the smooth manifold $\Sigma$ that the points…
user104605
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Is the closure of $ X \cap Y$ equal to $\bar{X} \cap \bar{Y}$?

$U$ is a topological space. $X$ is an open subset of $U$, and $Y$ is a closed subset. Let $Z = X \cap Y$. Does $\bar{Z} = \bar{X} \cap \bar{Y}$. Here, $\bar{X}$ denote the closure of $X$, and $\bar{Y}$, $\bar{Z}$ respectively. (So,…
ShinyaSakai
  • 7,846
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3 answers

$f : X \to Y $ continuous and surjective. $A $ dense in $X$ $\implies$ $f(A)$ dense in $Y$

$$ \textbf{PROBLEM} $$ $f : X \to Y $ continuous and surjective. $A $ dense in $X$ $\implies$ $f(A)$ dense in $Y$ $$ \textbf{SOLUTION(ATTEMPT)} $$ Suppose $A$ is dense in $X$. Then we must have by definition that $Cl(A) = X $. Since $f$ enjoys…
ILoveMath
  • 10,694
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Testing for topology on $\mathbb R^2$

I want to show that $\tau=\{G_k=(x,y):k\in \mathbb R\}\cup \{\mathbb{R}^2, \emptyset\}$, where $\forall k\in \mathbb R, \;\; G_k=\{(x,y):x>y+k\}$ is a topology on $\mathbb R^2$. My attemp: By definition $\mathbb R^2,\emptyset\in \tau$. Let…
user100943
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Find a point $l$ in the closure of $A$ so that no sequence with values in $A$ converges to $l$.

Find an example for this. You can choose which space to use. An example the professor gave had to do with the Box Topology, but I was wondering if there was an easier example. Thanks.
PitheMathemagician
  • 311
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Assuming every continuous function is uniformly continuous

If every continuous function, let's say $f:X\to \mathbb{R}$, that they are uniformly continuous, can I assume that $X$ is compact? I'm just wondering if I am true, can someone verify that I'm correct? Since compact of a set is defined by it being…
user97994
  • 179
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3 answers

Is the cartesian product of homeomorphisms again a homeomorphism?

If we have two homeomorphisms $f:A\to X$ and $g:B\to Y$, then is it true that $f\times g:A\times B\to X\times Y$ defined by $(f\times g)(a,b)=(f(a),g(b))$ is again a homeomorphism? I think the answer is yes; It's clearly a bijection. Intuitively it…
Xena
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