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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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If $f$ is continuous then $G$ is homeomorphic to $X$.

Let $f: X \to Y$ be a function. The graph of $f$ is defined to be the set $G = \{(x, f(x)) : x \in X\}$. Prove that if $f$ is continuous then $G$ is homeomorphic to $X$. an anyone suggest me the process of solving this problem, thanks.
atin
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Why do we care about the Hilbert Cube?

The Hilbert Cube is defined to be the countable infinite Cartesian products of the interval $[0,1]$ or anything homeomorphic to $[0,1]$. Why do we care about this object?
willyx888
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What conditions are sufficient for "Basically disconnectedness implies Extremally disconnectedness"?

Recall the definition of basically disconnected: A space is basically disconnected if every cozero-set has an open closure. There exists a Basically disconnected space which is not extremally disconnected; the one-point Lindelöfization of an…
TXC
  • 1,332
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What are sets made up of in topology?

I am looking for a very basic answer, I am in 10th grade and studying it for a project. I don't understand what the sets are referring to. Is it nodes? Edges? Axioms? Vertices? Or does it just depend?
hadley
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all topological spaces whose all subsets are compact

I have this question on my recent homework problem set, and the only thing I came up with, is to eliminate some "options", like metric spaces or compact spaces. Is there any characteristic answer for that? Is the answer as obviuos as just "finite…
Mahan Mehravard
  • 325
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Characterization of open maps

I'm looking for different but equivalent definitions of the concept of open map. So Let $X,Y$ be topological spaces and $f:X\longrightarrow Y$ a function, not assumed to be continuous. I conjectured the following equivalence: 1) $f$ is open, i.e.…
Federica Maggioni
  • 8,422
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If $X$ is a connected metric space, then a locally constant function $f: X \to $ M, $M $ a metric space, is constant

If $X$ is a connected metric space, then a locally constant function $f: X \to M$, $M $ a metric space, is constant. Thoughts: I can see that this is similar to the definition of connectedness: that any continuous map from $X$ to a two-point space…
user938272
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Does $\partial A$ determine $A$?

Given a bounded closed set $A$ in $\mathbb R^n$, can $A$ be uniquely determined by $\partial A$, except for the boundary itself? Or, use it differently, given two bounded closed sets $A_1, A_2$ in $\mathbb R^n$ with $\partial A_1 = \partial A_2$,…
Y choe
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Can the definition of "The Long Line" be clarified?

In Steen and Seebach's "Counterexamples in Topology", we see the definition of the Long Line (counterexample 45). "The long line $L$ is constructed from the ordinal space $[0, \Omega)$ (where $\Omega$ is the least uncountable ordinal) by placing…
Prime Mover
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Does Seperable + First Countable + Sigma-Locally Finite Basis Imply Second Countable?

A topological space is separable if it has a countable dense subset. A space is first countable if it has a countable basis at each point. It is second countable if there is a countable basis for the whole space. A collection of subsets of a…
Keshav Srinivasan
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Does this condition on a continuous surjection have a name?

Let $\pi : Y \to X$ be a continuous surjection of topological spaces. Does the following condition on $\pi$ have a name? For every $x \in X$, every open set $U$ containing $x$, and every $y \in \pi^{-1}(x)$, there exists a local section $s : U \to…
Qiaochu Yuan
  • 419,620
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$\mathbb{Z}$ and $\mathbb{Q}$ are not homeomorphic

Can anyone explain me why $\mathbb{Z}$ and $\mathbb{Q}$ are not homeomorphic? Thanks.
Muniain
  • 1,453
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Construct space $Y$ such that $Y\times X$ is extremally disconnected for any Tychonoff space $X$.

Let $X$ be a Tychonoff space. How can we construct a space $Y$ such that $Y\times X$ is extremally disconnected (Or Moscow) ?
TXC
  • 1,332
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When does boundedness imply totally boundedness in a metric space

For a subset of a metric space, quoted from Wikipedia: Total boundedness implies boundedness. For subsets of $\mathbb{R}^n$ the two are equivalent. I was wondering what are some more general conditions for a metric space than being…
Tim
  • 47,382
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A property between "separable" and "second countable"

Let $(X, \tau)$ be a topological space. It is second countable if it has a countable basis $B \subseteq \tau$. It is separable if there exists a countable $S \subseteq X$ such that $O \cap S \neq \emptyset$ for every nonempty $O \in \tau$. It is…
Doug McLellan
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