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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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Difference between Open sets of Interval and Open Sets of Topological Space

I am trying to understand the difference between open sets on a real line and open sets in a topological space. For example, while reading about open sets in Real line, it says: Recall the following definitions about open and closed sets in …
Wasiq Noor
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Is a subset of $\mathbb R^2$ homeomorphic to an open subset of $\mathbb R^2$ an open subset?

Suppose $X\subset \mathbb R^2$ with the subset topology and $U\subset\mathbb R^2$ an open subset with the subset topology, if $X\cong U$ and we deduce $X\subset \mathbb R^2$ is also open? This is not true for general topological space, for example…
Display Name
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Topology: Homeomorphism between finite complement topology in $\mathbb{R}$ and one of its subspaces

My class notes say that because $U=\mathbb{R}\backslash\{x_1,x_2,..,x_n\}$ has the same cardinality than $\mathbb{R}$, there exists a homeomorphism between: $(U,T_{cof})$ and $(\mathbb{R},T_{cof})$, where $T_{cof}$ is the finite complement…
Pao
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Characterization of discrete topology-like behavior about compact sets.

It's well-known that in discrete topology a set is compact iff it is finite. There exist a lot of examples of topologies which are not discrete but with that fact still holding, and it's not hard to find some of them. Is there any theory about…
Riccardo Plati
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Are there sets in the K-Topology that aren't open in the standard topology?

It seems to me that the basis for the K- Topology and the basis for the standard topology generate the same open sets. For instance, the open sets in the K-topology's basis that are different from the standard topology's basis are the sets which…
Akt904
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Finding a topology on $ \mathbb{R}^2 $ such that the $x$-axis is dense

The problem is the following Put a topology on $ \mathbb{R}^2$ with the property that the line $\{(x,0):x\in \mathbb{R}\}$ is dense in $\mathbb{R}^2$ My attempt If (a,b) is in $R^2$, then define an open sets of $(a,b)$ as the strip between $d$ and…
Miguel Mora Luna
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Topological basis vs local basis.

Given a topological space $(X, \tau)$ and a basis $B$ for $X$, every open set $A \in \tau$ can be written as $A = \cup B_\alpha$, where $B_\alpha \in B$. The definition of a local basis is supposed to be similar to this: If $B$ is a local basis for…
shivamvats
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show that $[0,1]$ is not compacts as a subspaces of $R_K$

Recall that $R_K$ denotes the R in the $K -topology$. a) Show that $[0,1]$ is not compacts as a subspaces of $R_K$ i know that $R_K $ is finer then R since its basis contain the basis of $R$ $R_K =(a,b) - \frac{1}{n}$ which is not…
jasmine
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two problems on compactness and connectedness of the space of matrices

Consider the set of all $n×n$ matrices with real entries as the space $\mathbb{R}^{n^2}$. Which of the following sets are compact? (a) The set of all orthogonal matrices. (b) The set of all matrices with determinant equal to unity. (c) The set of…
user55554
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fixed point of a map, closed unit disk to circle

Is it true that there exist $x\in S^1$ such that $f(x)=x$ where $f:D\rightarrow S^1$ is a continuous map. $D$ is closed unit disk.
Myshkin
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Is there any subset in $\mathbb{R^2}$ such that...

The intersection with every line L in $\mathbb{R^2}$ is open in $L$ with the topology $\mathcal{T}_L = \{G\cap L : G \text{ open in } \mathbb{R^2} \text{ with the Euclidean Topology } \} $, but the set is not open in $\mathbb{R^2}$ with the…
ChrisNick92
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Union of dense is dense?

Question:In topological space, union of any family of dense subset is dense? I don't know whether the above statement is true or not! I know the definition of dense sets in topological space. According to "me it may not be true, as closure of…
Akash Patalwanshi
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Taking a quotient of the 1-sphere by identifying diametrically opposite points

I have been working on the following problem: "Let $\sim$ be the equivalence relation on the unit circle $S^1$ defined by $x \sim -x$, $x \in S^1$. Show that $S^1/\sim$ is homeomorphic to $S^1$ and interpret geometrically." I have applied the…
Alex Petzke
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Quotient of the torus being homeomorphic to Mobius strip?

I'm considering X to be the space $[-1,1] \times [-1,1]$, with left glued to right and top glued to bottom, and points symmetric with respect to the diagonal glued to each other. But I still can't visualize the final shape of X. Is that a cone? And…
ensbana
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Understanding subspace topology

In wiki article of "compact spaces", they state that the set $\mathbb{Q} ∩ [0,1]$ is not compact because the sets of rational numbers in the intervals $[0, \frac{1}{π} - \frac{1}{n}]$ and $[ \frac{1}{π}+ \frac{1}{n}, 1]$ covers all the rationals in…
Akash Patalwanshi
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