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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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Elementary proof of topological invariance of dimension using Brouwer's fixed point and invariance of domain theorems?

http://people.math.sc.edu/howard/Notes/brouwer.pdf https://terrytao.wordpress.com/2011/06/13/brouwers-fixed-point-and-invariance-of-domain-theorems-and-hilberts-fifth-problem/ These papers give fairly elementary proofs of Brouwer's fixed point and…
Ormi
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10
votes
2 answers

one-point compactification of a compact space

We have the definition of the one-point compactification of a locally compact Hausdorff space: Let X be locally compact and Hausdorff and let Y be a compact Hausdorff space and $i:X\to Y$ such that there exists a $y_0\in Y:\;\; i:X\to Y\setminus…
user151465
10
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4 answers

homeomorphism non-example

A homeomorphism is a continuous function between topological spaces that has a continuous inverse function. Can someone provide examples of a continuous function between topological spaces that does not have a continuous inverse function? It would…
Alexandre Holden Daly
  • 203
10
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2 answers

Is it possible to obtain a sphere from a quotient of a torus?

I understand that a torus is obtained from a sphere by adding a handle. I'm working on a question which is asking if it is possible to obtain a sphere from a quotient of a torus? It seems like this should be possible by perhaps identifying the…
Wooster
  • 3,775
10
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1 answer

intersection of two curves in a square

I am sorry to ask this trivial question: Let $R$ be a square $[0,1]\times[0,1]$, and $A$ is a continuous curve from $(0,0)$ to $(1,1)$, while $B$ is another continuous curve from $(0,1)$ to $(1,0)$. Show that $A$ and $B$ always intersect. I have…
user18705
  • 103
10
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2 answers

Is there a countable, regular space with no isolated points which is not homeomorphic to the rationals?

It's known that every countable, metrizeable space with no isolated points is homeomorphic to the rationals with the standard topology. Suppose you wanted to reformulate the above without referencing metrizability directly. Since a countable space…
Mike F
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10
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2 answers

Limit points in topological space $X$

Let $X$ be a topological space. Let $A \subset X$. The point $x$ is a limit point of $A$ iff every neighborhood of $x$ contains a point $a$ of $A$ not equal to $x$. I am thinking about the following question: If $x$ is a limit point of $A$ does it…
blue
  • 2,884
10
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7 answers

Does homeomorphic to itself imply the same topology?

I know homeomorphim is an equivalence relation, which means a topological space will be homeomorphic to itself. However, does the converse hold? In other words, is it possible that a set with two different topologies can still be…
John
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10
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1 answer

In the Sorgenfrey plane, is an open disc homeomorphic to an open square?

In the sorgenfrey plane $\mathbb{R}_l^2$, the subspace $$X=\{(x,y):x^2+y^2\leq 1\}$$ is not homeomorphic to the subspace $$Y=\{(x,y):|x|\leq 1,|y|\leq 1\},$$ because there is only one isolated point in $Y$ but infinitely many in $X$. Now we modify…
Lili Shen
  • 103
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2 answers

Does closed imply bounded?

Definitions:1. A set $S$ in $\mathbb{R}^m$ is bounded if there exists a number $B$ such that $\mathbf{||x||}\leq B$ for all $\mathbf{x}\in S$, that is , if $S$ is contained in some ball in $\mathbb{R}^m$.2. A set in $\mathbb{R}^m$ is closed if,…
Silent
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2 answers

Connectedness of a certain subset of the plane

Let $U$ be an open and connected subspace of the Euclidean plane $\mathbb{R}^2$ and $A\subseteq U$ a subspace which is homeomorphic to the closed unit interval. Is $U\setminus A$ necessarily connected?
LostInMath
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10
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1 answer

Is a perfect set a boundary?

In a topological space, is a perfect set (i.e. closed with no isolated points) always the boundary of some set?
curious
  • 309
10
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1 answer

functions that send limit points to limit points

An exercise from Munkres states that: Suppose that $f: X \to Y$ is continuous. If $x$ is a limit point of a subset of $A$ of $X$, is it necessarily true that $f(x)$ is a limit point of $f(A)$? I think that the answer is no based on an example…
user96608
  • 103
10
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1 answer

Arbitrary product of simply connected space is simply connected?

Let $\lbrace X_\alpha\rbrace_{\alpha\in\Lambda}$ be a set of simply connected spaces. Is it true that $\pi_1(\prod\limits_{\alpha\in\Lambda} X_\alpha)=0$? cf) I know that $\pi_1(X\times Y)=\pi_1(X)\times\pi_1(Y)$.
Topologieeeee
  • 695
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2 answers

Pointwise topology embedding

First let $\Lambda$ be the bijective mapping between $Y^{Z \times X}$ and $(Y^X)^Z$ defined as follows: every mapping $f: Z \times X \to Y$ defines a set of mappings from $X$ to $Y$: for each $z \in Z$ is $f_z:X \to Y$ defined as $f_z(x) = f(z,x)$.…
JT_NL
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